1 university of waterloo department of mechanical engineering me 322 - mechanical design 1 partial...
TRANSCRIPT
1
University of Waterloo
Department of Mechanical Engineering
ME 322 - Mechanical Design 1ME 322 - Mechanical Design 1
Partial notes ndash Part 4 (Fatigue)(G Glinka)
Fall 2005
2
NNii = =
NNpp = =
NNTT = =
Metal FatigueMetal Fatigue is a process which causes premature irreversible damage or is a process which causes premature irreversible damage or failure of a component subjected to repeated loadingfailure of a component subjected to repeated loading
FATIGUE - What is itFATIGUE - What is it
3
Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several
disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics
bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component
bull Influenced by a componentrsquos environmentbull Takes many forms
ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue
Fatigue is not cause of failure per se but leads to the final fracture event
4
The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics
(from Ewalds amp Wanhil ref3)
5
Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
2
NNii = =
NNpp = =
NNTT = =
Metal FatigueMetal Fatigue is a process which causes premature irreversible damage or is a process which causes premature irreversible damage or failure of a component subjected to repeated loadingfailure of a component subjected to repeated loading
FATIGUE - What is itFATIGUE - What is it
3
Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several
disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics
bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component
bull Influenced by a componentrsquos environmentbull Takes many forms
ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue
Fatigue is not cause of failure per se but leads to the final fracture event
4
The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics
(from Ewalds amp Wanhil ref3)
5
Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
3
Metallic FatigueMetallic Fatiguebull A sequence of several very complex phenomena encompassing several
disciplinesndash motion of dislocationsndash surface phenomenandash fracture mechanicsndash stress analysisndash probability and statistics
bull Begins as an consequence of reversed plastic deformation within a single crystallite but ultimately may cause the destruction of the entire component
bull Influenced by a componentrsquos environmentbull Takes many forms
ndash fatigue at notchesndash rolling contact fatiguendash fretting fatiguendash corrosion fatiguendash creep-fatigue
Fatigue is not cause of failure per se but leads to the final fracture event
4
The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics
(from Ewalds amp Wanhil ref3)
5
Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
4
The Broad Field of Fracture MechanicsThe Broad Field of Fracture Mechanics
(from Ewalds amp Wanhil ref3)
5
Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
5
Intrusions and ExtrusionsIntrusions and Extrusions The Early Stages of Fatigue Crack Formation
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
6
Locally the crack grows in shear
macroscopically it grows in tension
Δσ
Δσ
c
Schematic of Fatigue Crack Initiation Subsequent Growth Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode ICorresponding and Transition From Mode II to Mode I
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
7
The Process of FatigueThe Process of Fatigue
The Materials Science PerspectiveThe Materials Science Perspective
bull Cyclic slipCyclic slipbull Fatigue crack initiationFatigue crack initiationbull Stage I fatigue crack growthStage I fatigue crack growthbull Stage II fatigue crack growthStage II fatigue crack growthbull Brittle fracture or ductile ruptureBrittle fracture or ductile rupture
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
8
Features of the Fatigue Fracture Surface of a Typical Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Ductile Metal Subjected to Variable Amplitude Cyclic
LoadingLoading
(Collins ref 22 )
A ndash fatigue crack area
B ndash area of the final static failure
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
9
Appearance of Failure Surfaces Caused by Appearance of Failure Surfaces Caused by Various Modes of LoadingVarious Modes of Loading (SAE Handbook)
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
10
Factors Influencing Fatigue LifeFactors Influencing Fatigue Life
Applied Stressesbull Stress range ndash The basic cause of plastic deformation and consequently the
accumulation of damagebull Mean stress ndash Tensile mean and residual stresses aid to the formation and
growth of fatigue cracksbull Stress gradients ndash Bending is a more favorable loading mode than axial
loading because in bending fatigue cracks propagate into the region of lower stresses
Materialsbull Tensile and yield strength ndash Higher strength materials resist plastic
deformation and hence have a higher fatigue strength at long lives Most ductile materials perform better at short lives
bull Quality of material ndash Metallurgical defects such as inclusions seams internal tears and segregated elements can initiate fatigue cracks
bull Temperature ndash Temperature usually changes the yield and tensile strength resulting in the change of fatigue resistance (high temperature decreases fatigue resistance)
bull Frequency (rate of straining) ndash At high frequencies the metal component may be self-heated
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
11
ComponentGeometry
LoadingHistory
Stress-StrainAnalysis
Damage Analysis
Allowable Load - Fatigue Life
MaterialProperties
Information path in strength and fatigue life prediction procedures
Strength-Fatigue Analysis ProcedureStrength-Fatigue Analysis Procedure
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
12
Stress Parameters Used in Static Strength and Stress Parameters Used in Static Strength and Fatigue AnalysesFatigue Analyses
peak
tn
peak
K
S
n
y
dn
0
T
peak
Str
ess
M
b)a)
n
y
dn
0
T
peak
Str
ess
S
S
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
13
Constant and Variable Amplitude Stress HistoriesConstant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
One cycle
m
max
min
Str
es
s
Time0
Constant amplitude stress historya)
Str
es
s
Time
0
Variable amplitude stress history
One reversal
b)
max min
max mi
max min
mi
max
n
n
2 2
2
peak peak peak
peak peak
peak peakpeak
m
peak
pe
peakpe
ak
aka
R
In the case of the peak stress history the important parameters are
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
14
Str
ess
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the ldquoRainflowrdquo Counted Cycles Stress History and the ldquoRainflowrdquo Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
15
A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation
1 1i i i iABS ABS
The stress amplitude of such a cycle is
1
2i i
a
ABS
The stress range of such a cycle is
1i iABS
The mean stress of such a cycle is
1
2i i
m
The Mathematics of the Cycle Rainflow Counting Method The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Historiesfor Fatigue Analysis of StressLoad Histories
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
16
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
17
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
18
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
19
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
20
Stress History
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Str
es
s (
MP
a)x
10
2
-3
-2
-1
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Reversal point No
Str
es
s (
MP
a)x
10
2
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
21
Number of Cycles Number of Cycles According to the According to the Rainflow Counting Rainflow Counting Procedure Procedure (N Dowling ref 2)
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
22
The FatigueThe Fatigue S-NS-N methodmethod (Nominal Stress Approach)
bull The principles of the S-N approach (the nominal stress method)
bull Fatigue damage accumulation
bull Significance of geometry (notches) and stress analysis in fatigue evaluations of engineering structures
bull
bull Fatigue life prediction in the design process
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
23
AB
Note In the case of smooth components such as the railway axle the nominal stress and the local peak stress are the same
peakS Smin
Smax
WWoumloumlhlerrsquos Fatigue Testhlerrsquos Fatigue Test
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
24
Characteristic parameters of the S - N curve areSe - fatigue limit corresponding to N = 1 or 2106 cycles for steels and N = 108 cycles for aluminum alloysS10
3 - fully reversed stress amplitude corresponding to N = 103 cyclesm - slope of the high cycle regime curve (Part 2)
Fully reversed axial S-N curve for AISI 4130 steel Note the break at the LCFHCF transition and the endurance limit
Number of cycles N
Str
ess
am
plitu
de
Sa (
ksi)
Sy
Se
Part 1 Part 2 Part 3
Infinite life
Su
S103
10m A maS C N N
Fatigue S-N curveFatigue S-N curve
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
25
Most of available S - N fatigue data has been obtained from fully reversed rotational bending tests However material behavior and the resultant S - N curves are different for different types of loading It concerns in particular the fatigue limit Se
The stress endurance limit Se of steels (at 106 cycles) and the fatigue strength S103
corresponding to 103 cycles for three types of loading can be approximated as (ref 1 23 24)
S103 = 090Su and Se = S106 = 05 Su - bending
S103 = 075Su and Se = S106 = 035 - 045Su - axial
S103 = 072Su and Se = S106 = 029 Su - torsion
103 104 105 106 10701
03
05
10
S103
Se
Number of cycles Log(N)
Rel
ativ
e st
ress
am
plitu
de
SaS
u
Torsion
AxialBending
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
26
Approximate endurance limit for various materialsApproximate endurance limit for various materials
Magnesium alloys (at 108 cycles) Se = 035Su
Copper alloys (at 108 cycles) 025Sult Se lt050Su
Nickel alloys (at 108 cycles) 035Su ltSe lt 050Su
Titanium alloys (at 107 cycles) 045Su ltSelt 065Su
Al alloys (at 5x108 cycles) Se = 045Su (if Su le 48 ksi) or Se = 19 ksi (if Sugt 48 ksi)
Steels (at 106 cycles) Se = 05Su (if Su le 200 ksi) or Se = 100 ksi (if Sugt200 ksi)
Irons (at 106 cycles) Se = 04Su (if Su le 60 ksi) or Se = 24 ksi (if Sugt 60 ksi)
S ndash N curveS ndash N curve
33
1 1 1
2
10101log lo
0
g
1
3
A
m mm ma
m m
e e
Aa aS C N or N C S C S
SSm and A
S S
N
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
27
Fatigue Limit ndash Modifying FactorsFatigue Limit ndash Modifying FactorsFor many years the emphasis of most fatigue testing was to gain
an empirical understanding of the effects of various factors on the base-line S-N curves for ferrous alloys in the intermediate to long life ranges The variables investigated include
- Rotational bending fatigue limit Sersquo
- Surface conditions ka
- Size kb
- Mode of loading kc Se = ka kb kc kd ke kfmiddotSersquo
- Temperature kd
- Reliability factor ke
- Miscellaneous effects (notch) kf
Fatigue limit of a machine part Se
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
28
Ca
Effect of various surface finishes on the fatigue limit of steel Shown are values of the ka the ratio of the fatigue limit to that for polished specimens
Below a generalized empirical graph is shown which can be used to estimate the effect of surface finish in comparison with mirror-polished specimens [Shigley (23) Juvinal (24) Bannantine (1) and other textbooks]
Surface Finish Effects on Fatigue Endurance LimitSurface Finish Effects on Fatigue Endurance LimitThe scratches pits and machining marks on the surface of a material add stress concentrations to the ones already present due to component geometry The correction factor for surface finish is sometimes presented on graphs that use a qualitative description of surface finish such as ldquopolishedrdquo or ldquomachinedrdquo
(from J Bannantine ref1)
k a
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
29
Size Effects on Endurance LimitSize Effects on Endurance LimitFatigue is controlled by the weakest link of the material with the probability of existence (or density) of a weak link increasing with material volume The size effect has been correlated with the thin layer of surface material subjected to 95 or more of the maximum surface stress
There are many empirical fits to the size effect data A fairly conservative one is
bull The size effect is seen mainly at very long lives
bull The effect is small in diameters up to 20 in (even in bending and torsion)
Stress effects in non-circular cross section members
In the case of non-circular members the approach is based on so called effective diameter de
The effective diameter de for non-circular cross sections is obtained by equating the volume of material stressed at and above 95 of the maximum stress to the same volume in the rotating-bending specimen
0097
0097
1 03
0869 03 100
10 8
1189 8 250
b
b
e
e
e
e
if d inS
S d if in d in
or
if d mmS
S d if d mm
k
k
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
30
+
-
095max
max
005d2
095dThe material volume subjected to stresses 095max is concentrated in the ring of 005d2 thick
The surface area of such a ring is
max
22 2095 095 00766
4A d d d
The effective diameter dThe effective diameter dee for members for members
with non-circular cross sectionswith non-circular cross sections
rectangular cross section under bending
F
095
t
t
Equivalent diameter
max095 095 005
A Ft
A Ft t Ft
200766 005
0808
e
e
d Ft
d Ft
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
31
Loading Effects on Endurance LimitLoading Effects on Endurance Limit
The ratio of endurance limits for a material found using axial and rotating bending tests ranges from 06 to 09
The ratio of endurance limits found using torsion and rotating bending tests ranges from 05 to 06 A theoretical value obtained from von Mises-Huber-Hencky failure criterion is been used as the most popular estimate
( ) ( )(07 09)
07 09 ( 085)c
e axial e bendi
c
ngS S
suggested by Shigl y kek
( ) ( )0577
057( 059)
e torsion e bendi
c
ng
c
S
suggested by Shigl kek y
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
32
From Shigley and Mischke Mechanical Engineering Design 2001
u T u T
e RT e RTu RT u R
dT
e T dSS S
S SS
kS
k
Temperature EffectTemperature Effect
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
33
Reliability factor Reliability factor kkee
The reliability factor accounts for the scatter of reference data such as the rotational bending fatigue limit Se
rsquo
The estimation of the reliability factor is based on the assumption that the scatter can be approximated by the normal statistical probability density distribution
1 008 ae zk The values of parameter za associated with various levels of reliability can be found in Table 7-7 in the textbook by Shigley etal
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
34
(source S Nishijima ref 39)
S-N curves for assigned probability of failure P - S - N curvesS-N curves for assigned probability of failure P - S - N curves
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
35
Stress concentration factor Stress concentration factor KKtt and the and the
notch factor effect notch factor effect kkff
Fatigue notch factor effect kf depends on the stress concentration factor Kt (geometry) scale and material properties and it is expressed in terms of the Fatigue Notch Factor Kf
1f
f
kK
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
36
1
peak
n
22
11
33
3
2
A D B
C
F
F
22
22
22
33
33
11
C
A B
D
Stresses in axisymmetric Stresses in axisymmetric notched bodynotched body
n
peakt n
FS
Aand
K
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
37
Stresses in prismatic notched bodyStresses in prismatic notched body
n
peakt n
t
FS
Aand
K
K S
A B C
22
22
22
33
D
E
11
peak
n
22
11
33
3
2
A D B
C
F
F
1
E
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
38
Stress concentration factors used in fatigue Stress concentration factors used in fatigue analysisanalysis
peak
tn
peak
K
S
n S
x
dn
0
W
peak
Str
ess
S
n
x
dn
0
W
Str
ess
M
peak
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
39
Stress concentration Stress concentration factors Kfactors Ktt in shafts in shafts
Bending load
Axial load
S =
S =
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
40
S =
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
41
Similarities and differences between the stress field near the notch Similarities and differences between the stress field near the notch and in a smooth specimenand in a smooth specimen
x
dn
0
W
peakS
tres
s
S
n=Speak= n=S
P
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
42
The Notch Effect in Terms of the NThe Notch Effect in Terms of the Nominal Stressominal Stress
np
n1
S2
cyclesN0N2
Sesmooth
N (S)m = C
Sm
ax
Str
ess
ran
ge
S
n2
n3
n4 notchedsmoothe
ef
SS
K
S
f tK KfK Fatigue notch factor
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
43
1
2
3
Nsm
ooth
22peak
smoothe
peak
Str
ess
1
2
Nnotched
notchede
M
notchedf notched
smoothsmoo he
e
tfo NNrK
Definition of the fatigue notch factor KDefinition of the fatigue notch factor Kff
1
22= peak
Str
ess
2
Nnotched
notchede
S
Se
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
44
11 1 1
1t
f t
KK q K
a r
1
1
qa r
PETERSONs approachPETERSONs approach
a ndash constant
r ndash notch tip radius
11
1t
f
KK
r
NEUBERrsquos approachNEUBERrsquos approach
ρ ndash constant
r ndash notch tip radius
18
330010
u
u
a inS
for S in in
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
45
The Neuber constant The Neuber constant lsquolsquoρρrsquorsquo for steels and aluminium alloys for steels and aluminium alloys
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
46
Curves of notch sensitivity index lsquoqrsquo versus notch radiusCurves of notch sensitivity index lsquoqrsquo versus notch radius (McGraw Hill Book Co from ref 1)
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
47
Plate 1
W1 = 50 in
d1 = 05 in
Su = 100 ksi
Kt = 27
q = 097
Kf1 = 265
Illustration of the notchscale effectIllustration of the notchscale effect
Plate 2
W2= 05 in
d2 = 05 i
Su = 100 ksi
Kt = 27
q = 078
Kf1 = 232
peak
m1
d1
W1
peak
d2
W2
m2
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
48
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Nf (logartmic)
JuvinalShigley method
Nf (logartmic)
Collins method
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
frsquo
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
49
Procedures for construction of approximate fully reversedProcedures for construction of approximate fully reversed S-NS-N curves for smooth and notched componentscurves for smooth and notched components
Sar ar ndash nominallocal stress amplitude at zero mean stress m=0 (fully reversed cycle)
Sa
r (lo
gar
tmic
)
Nf (logartmic)
Manson method09Su
100 101 102 103 104 105 106 2106
Sersquokakckbkdke
Sersquokakckbkdkekf
Se
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
50
NOTENOTE
bull The empirical relationships concerning the S ndashN curve data are only estimates Depending on the acceptable level of uncertainty in the fatigue design actual test data may be necessary
bull The most useful concept of the S - N method is the endurance limit which is used in ldquoinfinite-liferdquo or ldquosafe stressrdquo design philosophy
bull In general the S ndash N approach should not be used to estimate lives below 1000 cycles (N lt 1000)
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
51
Constant amplitude cyclic stress historiesConstant amplitude cyclic stress histories
Fully reversedFully reversed
mm = 0 R = -1 = 0 R = -1
PulsatingPulsating
mm = = aa R = 0 R = 0
CyclicCyclic
mm gt 0 R gt 0 gt 0 R gt 0
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
52
Mean Stress EffectMean Stress Effect
smlt 0
sm= 0
smgt 0
No of cycles logN 2106
Str
ess
ampl
itude
log
Sa
time
smlt 0
sm= 0
smgt 0
Str
ess
ampl
itude
Sa
0
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
53
The tensile mean stress is in general detrimental while the compressive mean stress is beneficial or has negligible effect on the fatigue durability
Because most of the S ndash N data used in analyses was produced under zero mean stress (R = -1) therefore it is necessary to translate cycles with non- zero mean stress into equivalent cycles with zero mean stress producing the same fatigue life
There are several empirical methods used in practice
The Hiagh diagram was one of the first concepts where the mean stress effect could be accounted for The procedure is based on a family of Sa ndash Sm curves obtained for various fatigue lives
Steel AISI 4340 Steel AISI 4340
SSyy = 147 ksi = 147 ksi (Collins)
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
54
MeanMean Stress Correction for Endurance Limit Stress Correction for Endurance Limit
Gereber (1874)S
S
S
Sa
e
m
u
2
1
Goodman (1899)S
S
S
Sa
e
m
u
1
Soderberg (1930)S
S
S
Sa
e
m
y
1
Morrow (1960)S
S
Sa
e
m
f
1
Sa ndash stress amplitude applied at the mean stress Smne 0 and fatigue life N = 1-2x106cycles
Sm- mean stress
Se- fatigue limit at Sm=0
Su- ultimate strength
frsquo- true stress at fracture
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
55
MeanMean stress correction for arbitrary stress stress correction for arbitrary stress amplitude applied at non-zero mean stressamplitude applied at non-zero mean stress
Gereber (1874)
2
1a m
ar u
S S
S S
Goodman (1899) 1a m
ar u
S S
S S
Soderberg (1930) 1a m
ar y
S S
S S
Morrow (1960)
1a m
ar f
S S
S
Sa ndash stress amplitude applied at the mean stress Smne 0 and resulting in fatigue life of N cycles
Sm- mean stress
Sar- fully reversed stress amplitude applied at mean stress Sm=0 and resulting in the same fatigue life of N cycles
Su- ultimate strength
frsquo- true stress at fracture
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
56
Comparison of various Comparison of various methods of accounting methods of accounting for the mean stress effectfor the mean stress effect
Most of the experimental data lies between the Goodman and the yield line
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
57
Approximate Goodmanrsquos diagrams for ductile Approximate Goodmanrsquos diagrams for ductile and brittle materialsand brittle materials
Kf
Kf
Kf
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
58
The following generalisations can be made when discussing mean stress effects
1 The Soumlderberg method is very conservative and seldom used
3 Actual test data tend to fall between the Goodman and Gerber curves
3 For hard steels (ie brittle) where the ultimate strength approaches the true fracture stress the Morrow and Goodman lines are essentially the same For ductile steels (of gt S) the Morrow line predicts less sensitivity to mean stress
4 For most fatigue design situations R lt 1 (ie small mean stress in relation to alternating stress) there is little difference in the theories
5 In the range where the theories show a large difference (ie R values approaching 1) there is little experimental data In this region the yield criterion may set design limits
6 The mean stress correction methods have been developed mainly for the cases of tensile mean stress
For finite-life calculations the endurance limit in any of the equations can be For finite-life calculations the endurance limit in any of the equations can be replaced with a fully reversed alternating stress level corresponding to that replaced with a fully reversed alternating stress level corresponding to that finite-life valuefinite-life value
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
59
Procedure for Fatigue Damage CalculationProcedure for Fatigue Damage Calculation
NT
n1
n2
n3
n4
cycles
e
NoN2
e
Ni(i)m = a
N1 N3
Str
ess
rang
e
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cycles applied atn cycles applied at n cycles applied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
1 1 2 2
1 1
Ri i
LD n N n N n N
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
60
n1 - number of cycles of stress range 1
n2 - number of cycles of stress range 2 ni - number of cycles of stress range i
11
1D
N - damage induced by one cycle of stress range 1
11
1n
nD
N - damage induced by n1 cycles of stress range 1
22
1D
N - damage induced by one cycle of stress range 2
22
2n
nD
N - damage induced by n2 cycles of stress range 2
1i
i
DN
- damage induced by one cycle of stress range i
ini
i
nD
N - damage induced by ni cycles of stress range i
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
61
Total Damage Induced by the Stress HistoryTotal Damage Induced by the Stress History
1 1 2 2
1 1 2 2
1 21 2
11 2
i i
i i
ii i
n n nii i
n cyclesapplied atn cyclesapplied at n cyclesapplied atD
N cycles to failure at N cycles to failure at N cycles to failure at
n nn nD D D D
N N N N
It is usually assumed that fatigue failure occurs when the cumulative damage exceeds some critical value such as D =1
ie if D gt 1 - fatigue failure occurs
For D lt 1 we can determine the remaining fatigue life
1 1 2 2
1 1
Ri i
LD n N n N n N
LR - number of repetitions of the stress history to failure
1 2 3 R iN L n n n n N - total number of cycles to failure
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
62
jN = am
j
if j gt e
It is assumed that stress cycles lower than the fatigue limit j lt e produce no damage (Nj=) in the case of constant amplitude loading however in the case of variable amplitude loading the extension of the S-N curve with the slope lsquom+2rdquo is recommended The total damage produced by the entire stress spectrum is equal to
1
j
jj
D D
It is assumed that the component fails if the damage is equal to or exceeds unity ie when D 1 This may happen after a certain number of repetitions BL (blocks) of the stress spectrum which can be calculated as
BL = 1D
Hence the fatigue life of a component in cycles can be calculated as
N = BLNT where NT is the spectrum volume or the number of cycles extracted from given stress history
NT = (NOP - 1)2
If the record time of the stress history or the stress spectrum is equal to Tr the fatigue life can be expressed in working hours as
T = BL Tr
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
63
Main Steps in the Main Steps in the S-NS-N Fatigue Life Estimation Procedure Fatigue Life Estimation Procedure
bull Analysis of external forces acting on the structure and the component in question
bull Analysis of internal loads in chosen cross section of a componentbull Selection of individual notched component in the structurebull Selection (from ready made family of S-N curves) or construction of S-
N curve adequate for given notched element (corrected for all effects)bull Identification of the stress parameter used for the determination of the
S-N curve (nominalreference stress)bull Determination of analogous stress parameter for the actual element in
the structure as described abovebull Identification of appropriate stress historybull Extraction of stress cycles (rainflow counting) from the stress historybull Calculation of fatigue damagebull Fatigue damage summation (Miner- Palmgren hypothesis)bull Determination of fatigue life in terms of number of stress history
repetitions Nblck (No of blocks) or the number of cycles to failure Nbull The procedure has to be repeated several times if multiple stress
concentrations or critical locations are found in a component or structure
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
64
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
65
Stress History
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 5 10 15 20 25 30 35
Reversal point No
Str
ess
(ksi
)
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
66
411108861
log60log6
log110log3
log60log10log
log110log1000log
logloglog
26
6
mC
Cm
Cm
Cm
Cm
CSmN
CSN
a
ma
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
67
S-N Curve
10
100
1000
1000 10000 100000 1000000 10000000
No of cycles
Str
es
s a
mp
litu
de
(k
si)
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
68
Goodman Diagram
0
10
20
30
40
50
60
70
80
90
100
-100 -50 0 50 100 150
Mean stress (ksi)
Str
es
s a
mp
litu
de
(k
si)
SY
Suts
1
0
0
1 1
11
m
m
a m maa r at S
e uts uts
a m aa r at S
ma r uts
uts
S S Sfor fatigue endurance S S
S S S
S S Sfor any stress amplitude S
SS SS
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
69
Calculations of Fatigue Damage a) Cycle No11
Sar=8793 ksi
N11= C( SaN)-m= 186610268793-114=12805 cycles
D11=0000078093 b) Cycle No 14
Sar=750 ksi
N14= C( SaN)-m= 18661026750-114=78561 cycles
D14=0000012729 c) Cycle no 15
Sar=9828 ksi
N14= C( SaN)-m= 186610269828-114=3606 cycles
D14=000027732
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-
70
Results of rainflow counting D a m a g eNo S Sm Sa Sar (Sm=0) Di=1Ni=1CSa
-m
1 30 5 15 1552 20155E-13 02 40 30 20 2500 46303E-11 03 30 45 15 2143 79875E-12 04 20 -50 10 1000 13461E-15 05 50 -45 25 2500 46303E-11 06 30 5 15 1552 20155E-13 07 100 20 50 5769 63949E-07 08 20 -20 10 1000 13461E-15 09 30 -25 15 1500 13694E-13 010 30 -55 15 1500 13694E-13 011 170 5 85 8793 78039E-05 780E-0512 30 -5 15 1500 13694E-13 013 30 -5 15 1500 13694E-13 014 140 10 70 7500 12729E-05 127E-0515 190 5 95 9828 000027732 0000277
n0=15 D = 000036873 3677E-04
D=0000370 D=3677E-04 LR = 1D =271203 LR = 1D =271961
N=n0LR=15271203=40680 N=n0LR=15271961=40794
- Slide 1
- Slide 2
- Metallic Fatigue
- The Broad Field of Fracture Mechanics
- Intrusions and Extrusions The Early Stages of Fatigue Crack Formation
- Schematic of Fatigue Crack Initiation Subsequent Growth Corresponding and Transition From Mode II to Mode I
- The Process of Fatigue
- Features of the Fatigue Fracture Surface of a Typical Ductile Metal Subjected to Variable Amplitude Cyclic Loading
- Appearance of Failure Surfaces Caused by Various Modes of Loading (SAE Handbook)
- Factors Influencing Fatigue Life
- Strength-Fatigue Analysis Procedure
- Stress Parameters Used in Static Strength and Fatigue Analyses
- Constant and Variable Amplitude Stress Histories Definition of a Stress Cycle amp Stress Reversal
- Stress History and the ldquoRainflowrdquo Counted Cycles
- The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of StressLoad Histories
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- The Fatigue S-N method (Nominal Stress Approach)
- Woumlhlerrsquos Fatigue Test
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- Slide 39
- Slide 40
- Slide 41
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Slide 47
- Procedures for construction of approximate fully reversed S-N curves for smooth and notched components
- Slide 49
- Slide 50
- Slide 51
- Slide 52
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Slide 64
- Slide 65
- Slide 66
- Slide 67
- Slide 68
- Slide 69
- Slide 70
-