Fatigue Failure Theories
Modified by Paul Meyer
Fatigue
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To this point
– Static loads
– Static stresses
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Is this the case for mechanical design?
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Can one apply a load that is less than the yield strength and still
induce failure?
2
Modified by Paul Meyer
Fatigue
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History
–
Introduction of engines in 1800’s
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–
August Wohler (1870) studied axle failure
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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
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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)
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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
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Cracks
–
–
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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
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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
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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”
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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.
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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 )
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Why do each of these things factor into our adjustment?
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Let us consider each individually.
15
Modified by Paul Meyer
Se` and Sf’
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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)
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Csize = Kb
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For circular cross sections
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18
NOTE: the accuracy of these equations for nonferrous
materials is questionable
Modified by Paul Meyer
Size Correction
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Csize
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For non-circular cross sections – adjust using
A95
d eq=
0.0766
19
Modified by Paul Meyer
Size Correction
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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
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Kf
–
Corrosion
Effect of Fresh Water
Effect of Environment
Effect of Saltwater
25
Modified by Paul Meyer
Miscellaneous Effects Factor
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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
⚫
⚫
⚫
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⚫
⚫
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
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Notch:
– Any geometric characteristic that disrupts the “force flow”
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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
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Theoretical Stress Concentrations
– Kt (see Appendices or other books for values) – only
geometric
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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
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30
Neuber’s Constant for aluminums
Modified by Paul Meyer
Stress Concentrations
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31
Notch sensitivity – q for bending or axial loading
Modified by Paul Meyer
Stress Concentrations
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Notch sensitivity – q for reverse torsional loading
Modified by Paul Meyer
Application of Kf for Fluctuating Stresses
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For fluctuating loads at points with stress concentration, the best
approach is to design to avoid all localized plastic strain.
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In this case, Kf should be applied to both alternating and
midrange stress components.
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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
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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
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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
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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
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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