Total Strain Amplitude (Manson-Coffin Relationship)
Δ𝜖 𝜎𝐹′
= (2𝑁)𝑏 + 𝜖𝐹′ (2𝑁)𝑐
2
𝐸
LEFM Crack growth, Stress Intensity Factor
Endurance Limit Modifying Factors:
Temperature Factor, 𝑘𝑑
𝑆𝑒 = 𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘𝑑 𝑘𝑒 𝑘𝑓 𝑆𝑒′
𝑘𝑑 =
𝑏
Surface Modification Factor, 𝑘𝑎 = 𝑎𝑆𝑢𝑡
Stress Concentration (𝑲𝒕 ) and Notch Sensitivity (𝒒)
𝑆𝑇
𝑆𝑅𝑇
Under axial/bending loads
Δ𝐾𝐼 = 𝛽Δ𝜎√𝜋𝑎
Paris Equation:
𝑑𝑎
𝑑𝑁
Some materials are less sensitive to notches, thus the
use of 𝐾𝑓 in place of 𝐾𝑡
𝑞=
= 𝐶(Δ𝐾𝐼 )𝑚
𝐾𝑓 − 1
=
𝐾𝑡 − 1
1
1+
√𝑎
√𝑟
Under shear loading
𝑞𝑠ℎ𝑒𝑎𝑟 =
Size Factor, 𝑘𝑏
Endurance Limit, 𝑆𝑒′ , variation with tensile strength, 𝑆𝑢𝑡
For rotating bars, circular cross-section, under bending
and torsion
0.88𝑑−0.107
−0.157
= { 0.91𝑑−0.107
1.24𝑑
1.51𝑑−0.157
0.11 ≤ 𝑑 ≤ 2 in
𝑑 < 𝑑 ≤ 10 in
2.79 ≤ 𝑑 ≤ 51 mm
51 < 𝑑 ≤ 254 mm
For rotating bars, circular cross-section, under axial
loading, 𝑘𝑏 = 1
For non-rotating bars with non-circular cross-section,
the equivalent diameter 𝑑𝑒 is used
𝐾𝑓𝑠 − 1
𝐾𝑡𝑠 − 1
0≤𝑞≤1
If the endurance limit is not known, then
For fully reversed bending and axial loads:
𝑘𝑑 = 0.975 + 0.432(10−3 )𝑇𝐹 − 0.115(10−5 )𝑇𝐹2
+ 0.104(10−8)𝑇𝐹3 − 0.595(10−12 )𝑇𝐹4
for 70℉ ≤ 𝑇𝐹 ≤ 1000℉
Reliability Factor, 𝑘𝑒
𝑘𝑒 = 1 − 0.08𝑧𝑎
Fatigue strength factor at 103 cycles:
(𝑆𝑓′ )
103
= 𝜎𝐹′ (2 × 103 )𝑏 = 𝑓𝑆𝑢𝑡
2
√𝑎 = 0.246 − 3.08(10−3 )𝑆𝑢𝑡 + 1.51(10−5 )𝑆𝑢𝑡
3
− 2.67(10−8)𝑆𝑢𝑡
In general (S-N relation), (𝑆𝑓′ ) = 𝜎𝐹′ (2𝑁)𝑏
𝑁
Miscellaneous-Effects Factor, 𝑘𝑓
For fully reversed torsional loads:
Loading Factor, 𝑘𝑐
Alternatively, 𝑆𝑓 = 𝑎𝑁 𝑏
(𝑓𝑆𝑢𝑡 )2
𝑎=
,
𝑆𝑒
1
𝑓𝑆𝑢𝑡
𝑏 = − log (
)
3
𝑆𝑒
1.00 bending
= {0.85 axial
0.59 torsion
Failure may occur not at the outermost surface but at
the interface between the core and the case, where
stress exceeds the core's endurance limit.
2
√𝑎 = 0.190 − 2.51(10−3 )𝑆𝑢𝑡 + 1.35(10−5 )𝑆𝑢𝑡
3
− 2.67(10−8)𝑆𝑢𝑡
Characterization of Fluctuating Stress
Fatigue Diagram with various criteria of failure
Intersecting Coordinates in the First Quadrant for
Modified Goodman and Langer Failure Criteria
Intersecting Coordinates in the First Quadrant for
ASME-Elliptic and Langer Failure Criteria
Intersecting Coordinates in the First Quadrant for
Gerber and Langer Failure Criteria
Combination of Loading Modes
Soderger
𝜎𝑎 𝜎𝑚 1
+
=
𝑆𝑒 𝑆𝑦 𝑛
𝑅=
𝜎𝑚𝑖𝑛
,
𝜎𝑚𝑎𝑥
𝐴=
𝜎𝑎
𝜎𝑚
A sample master fatigue diagram for AISI4340 steel
having 𝑆𝑢𝑡 = 158 kpsi and 𝑆𝑦 = 147 kpsi. The stress
components at point A are 𝜎𝑚𝑖𝑛 = 20 kpsi, 𝜎𝑚𝑎𝑥 =
120 kpsi, 𝜎𝑚 = 70 kpsi, and 𝜎𝑎 = 50 kpsi
Mod-Goodman
𝜎𝑎 𝜎𝑚 1
+
=
𝑆𝑒 𝑆𝑢𝑡 𝑛
Gerber
𝑛𝜎𝑎
𝑛𝜎𝑚 2
+(
) =1
𝑆𝑒
𝑆𝑢𝑡
Distortion energy failure is the most reliable theory to
be used.
𝜎𝑎′ = {[(𝐾𝑓 )𝑏𝑒𝑛𝑑𝑖𝑛𝑔 (𝜎𝑎 )𝑏𝑒𝑛𝑑𝑖𝑛𝑔
ASME-Elliptic
2
𝑛𝜎𝑎 2
𝑛𝜎𝑚
(
) +(
) =1
𝑆𝑒
𝑆𝑦
Langer Static Yield
2
(𝜎𝑎 )𝑎𝑥𝑖𝑎𝑙
+ (𝐾𝑓 )𝑎𝑥𝑖𝑎𝑙
]
0.85
2
𝜎𝑎 𝜎𝑚 1
+
=
𝑆𝑦 𝑆𝑦 𝑛
n: Design factor or the factor of safety
1/2
+ 3 [(𝐾𝑓𝑠 )𝑡𝑜𝑟𝑠𝑖𝑜𝑛 (𝜏𝑎 )𝑡𝑜𝑟𝑠𝑖𝑜𝑛 ] }
′
𝜎𝑚
= {[(𝐾𝑓 )𝑏𝑒𝑛𝑑𝑖𝑛𝑔 (𝜎𝑚 )𝑏𝑒𝑛𝑑𝑖𝑛𝑔
2
+ (𝐾𝑓 )𝑎𝑥𝑖𝑎𝑙 (𝜎𝑚 )𝑎𝑥𝑖𝑎𝑙 ]
Load Line
2 1/2
𝑟=
𝜎𝑎
𝑆𝑎
=
𝜎𝑚 𝑆𝑚
+ 3 [(𝐾𝑓𝑠 )𝑡𝑜𝑟𝑠𝑖𝑜𝑛 (𝜏𝑚 )𝑡𝑜𝑟𝑠𝑖𝑜𝑛 ] }
When using failure criteria for multi-axial loads, these
formulations can be used directly. While calculating
modified 𝑆𝑒 , no need to account 𝑘𝑐 in this formulation
as it has already been considered here.