Advances in fatigue and fracture mechanics by grzegorz (greg) glinka

© 2014 Grzegorz Glinka. All rights reserved. 1
Advances in Fatigue and Fracture Mechanics
June 2nd – 6th, 2014,
Aalto University,
Espoo, Finland
given by
Prof. Darrell Socie
University of Illinois, USA
Prof. Grzegorz Glinka
University of Waterloo, Canada

1. Bannantine, J., Corner, Handrock, Fundamentals of Metal Fatigue Analysis, Prentice-Hall,
1990.... (good general reference)
2. Dowling, N., Mechanical Behaviour of Materials, Prentice Hall, 2011, 3rd edition
(middle chapters are a great overview of most recent approaches to fatigue analysis),
3. Stephens, R.I., Fatemi, A., Stephens, A.A., Fuchs, H.O., Metal Fatigue in Engineering, John
Wiley, 2001.... (good general reference),
4. M. Janssen, J. Zudeima, R.J.H. Wanhill, Fracture Mechanics, VSSD, The Netherlands, 2006
(understandable, rigorous, mechanics perspective),
5. Socie, D.F., and Marquis, G.B., Multiaxial Fatigue, Society of Automotive Engineers, Inc.,
Warrendale, PA, 2000
5. Haibach, E., Betriebsfestigkeit, VDI Verlag, Dusseldorf, 1989 (in German).
6. Bathias, C., and Pineau, A., Fatigue des Materiaux et des Structures, Hermes, Paris, 2008 (in
French and English),
7. Radaj, D., Design and Analysis of Fatigue Resistant Structures, Halsted Press, 1990,
(Complete, civil and automotive engineering analysis perspective),
8. V.A. Ryakhin and G.N. Moshkarev, Durability and Stability of Welded Structures in Earth Moving
Machinery”, Mashinostroenie, Moscow, 1984 (in Russian, cranes and earth moving machinery),
9. A. Chattopadhyay, G. Glinka, M. El-Zein, J. Qian and R. Formas, Stress Analysis and Fatigue of
Welded Structures, Welding in the World, (IIW), vol. 55, No. 7-8, 2011, pp. 2-21.
Bibliography

Mechanical Engineer – yesterday and today……
Slide ruler Calculator Computer PC/laptop
1-2 operation/min. 1-10 operation/min. 10? operation/min.
Before yesterday Yesterday Today

DAY 1
Contemporary Fatigue Analysis
Methods
(basics concepts and assumptions)

Information Path for Strength and Fatigue Life Analysis
Component
Geometry
Loading
History
Stress-Strain
Analysis
Damage Analysis
Fatigue Life
Material
Properties

Stress Parameters Used in Fatigue Analyses
K YS a
a)
Sn
y
dn
0
T
e
peak
S
y
dn
0
T
peak
M
b) c)
y
a
0
T
K
a2
S
S
a
peak
e
peak
Sn crack
Sn – net nominal stress; S – gross nominal stress
e
peak – local linear-elastic notch-tip stress
a
peak – local actual elastic-plastic notch-tip stress
Kt = e
peak /Sn – stress concentration factor
K – stress intensity factor

What stress parameter is needed for the Fracture
Mechanics based (da/dN- K) fatigue analysis?
x
a
0
T
Stress(x)
2
x
x
K
S
S
The Stress Intensity Factor K characterizing
the stress field in the crack tip region is
needed!
The K factor can be obtained from:
- ready made Handbook solutions (easy to use
but often inadequate in practice)
- from the near crack tip stress (x)
distribution or the displacement data obtained
from FE analysis of a cracked body (tedious)
- from the weight function by using the FE
stress analysis data of un-cracked body
(versatile and suitable for FCG analysis)

Loads and stresses in a structure
n
Load F
peak

The Most Popular Methods for
Fatigue Life Analysis - outlines
• Stress-Life Method or the S - N approach;
uses the nominal or simple engineering stress ‘ S ‘ to
quantify fatigue damage
• Strain-Life Method or the - N approach;
uses the local notch tip strains and stresses to quantify
the fatigue damage
• Fracture Mechanics or the da/dN - K approach;
uses the stress intensity factor to quatify the fatigue
crack growth rate

Information path for fatigue life estimation based on
the S-N method
LOADING
F
t
GEOMETRY, Kf
PSO
MATERIAL
0
E
Stress-Strain
Analysis
Damage Analysis
Fatigue Life
MATERIAL
No N
n
e
A
D
F

a) Structure
Q
H
F
K 5
n
The Similitude Concept states that if the nominal stress histories in the structure and in the test
specimen are the same, then the fatigue response in each case will also be the same and can be
described by the generic S-N curve. It is assumed that such an approach accounts also for the stress
concentration, loading sequence effects, manufacturing etc.
K0
K1
K2
K3
K4
K5
Stressamplitude,n/2orhs/2
Number of cycles, N
0
The Similitude Concept in the S-N Method

Steps in Fatigue Life Prediction Procedure Based on the
S-N Approach
The S – N method
5
e) Standard S-N curves
K0
K1
K2
K3
K4
K5
Stressamplitude,n/2orhs/2
Number of cycles, N
d) Standard welded joints
c) Section with welded joint
a) Structure
b) Component
V
P
R
Q
H
F
Weld

5
5
5
5
5
1
1
1 5
2
2
2 5
3
3
3 5
4
4
4 5
5
5
5 5
1
1
1
1
1
m
m
m
m
m
D
N C
D
N C
D
N C
D
N C
D
N C
1 2 3 4 5 ;D D D D D D
1
2
3 4
5
Stress
t
g)
Fatigue damage:h)
Total damage:i)
Fatigue life: N blck=1/Dj)
Stress,n
f)
t
K5
n
Steps in Fatigue Life Prediction Procedure Based
on the S-N Approach (continued)

The linear hypothesis of Fatigue Damage Accumulation
(the Miner rule)
1
1
1
1
;
m
D
N A
31 52 4
1 1
;
1 1 1 1 1R RRfL
N
L
N
N
N
n
D NN
2
2
2
1
;
m
D
N A
4
4
4
1
;
m
D
N A
5
1
0;D 42 51 3
5
5
1
32 41
1 1 1 1 1
;
1 !!
i
i
D
N
D D
N
D
N
D D
NN
D
if D Failure
e
N0N2N1 N3
Stressrange,
N4
N5
Cycles
Ni( i)m = A or Ni=A/( i)m
nR
3
3
3
1
;
m
D
N A

The FALSN fatigue life estimation software – Typical input and output data
Weldment

The scatter in fatigue:Fatigue S-N curves for assigned probability
of failure; P-S-N curvesStressamplitudeSa[N/mm2]
Number of cycles N
400
450
500
105
106 107
(source: S. Nishijima, ref. 39)
S45 Steel tempered
at 600o C (W1)
Probability of failure
P(%)
P=99%
P=90%
P=50%
P=10%
P=1%
8
6
48
108
f(N)
f(S)
m
a A N

N
j < L
T
COMPUTING of Tj
SAMPLING A SET OF RANDOM DATA
fK
KfK
fR
RwR
fS
SmaxS
STRUCTURAL COMPONENT
PWL
MATERIAL
No N
S
Rw
LOADING
S
t
Failure probability calculation
Pf = P[T(X) Tr]:
( )f
f
L T Tr
P
L
Probabilistic fatigue life assessment

Over designedUn-satisfactory Most
frequent
Characteristic regions of cumulative
probability of the fatigue life distribution

LOADING
F
t
GEOMETRY, Kf
PSO
MATERIAL
0
E
Stress-Strain
Analysis
Damage Analysis
Fatigue Life
Information path for fatigue life
estimation based on the -N method
MATERIAL
2Nf

a) Specimen
b) Notched component
peak
x
y
z
peak
peak
'
'
2 2
2
bf
f f fN N c
E
log(/2)
log
(2Nf)
f
0
l)
j)
The Similitude Concept states that if
the local notch-tip strain history in the
notch tip and the strain history in the
test specimen are the same, then the
fatigue response in the notch tip region
and in the specimen will also be the
same and can be described by the
material strain-life ( -N) curve.
The Similitude Concept in the – N Method

Steps in fatigue life prediction procedure based on
the - N approach
a) Structure
b) Component
d)
peak
n
peak
hs
nhs

Fatigue damage:
1 2 3 4
1 2 3 4
1 1 1 1
; ; ; ;D D D D
N N N N
Total damage:
1 2 3 4 ;D D D D D
Fatigue life: N blck=1/D
e)
log(/2)
'
'
2 2
2
b cf
f f fN N
E
e/E
log(2Nf)
f/E
f
0
2Ne
2N
2
:
peak
Neuber
E
' '
' '
'
'
,
,
,
?,?
f f
f f
f
f
fN
p e=
peak
t
1
2
3
4
5
6
7
8
1'
f)
0
t
'
1
'
n
E K
0
3
2,2'
4
5,5'
7,7'
6
8
1,1'
(continued). Steps in fatigue life prediction procedure based on the -N approach

Three basic sets of input data for the evaluation of the Fatigue Crack
Initiation Life and Reliability (the – N approach)
T T
R(T) = 1 - F(T)
1
f(T)
LOADING
S
t
f(k)
Scaling factor k
LIFE CALCULATION: Ti
i < L
Y
SAMPLING: k , Kt , f , f' , K’
N
COMPONENT
f(Kt)
SCF Kt
K5
S S
Computing of failure probabilities
( )f r
f r
L T T
P P T X T
L
MATERIAL
2N 1
K'
fM1
’fe
fM3
K’M
fM2
’fs
2p
2,
f
,
f
E
2

Probabilityoffailure

Information path for fatigue life estimation based on
the da/dN- K method
LOADING
F
t
GEOMETRY, Kt
PSO
MATERIAL
0
E
Stress-Strain
Analysis
Damage Analysis
Fatigue Life
MATERIAL
n
KKth
da
dN

The Similitude Concept states that if the stress
intensity K for a crack in the actual component
and in the test specimen are the same, then the
fatigue crack growth response in the component
and in the specimen will also be the same and
can be described by the material fatigue crack
growth curve da/dN - K.
a) Structure
Q
H
F
a
b) Weld detail
c) Specimena
P
P
K
10-12
10-11
10-10
10-9
10-8
10-7
10-6
1 10 100
CrackGrowthRate,m/cycle
mMPa,K
The Similitude Concept in the da/dN – K Method

V
P
Q
R
H
F
Weld
a) Structure
b) Component
S1
S2
S3
S4
S5
Stress,S
t
e)
d)
Steps in the Fatigue Life Prediction Procedure Based
on the da/dN- K Approach

Stress intensity factor, K
(indirect method)
Weight function, m(x,y)
, ,
A
n
K x y m x y dxdy
K
Y
a
Stress intensity factor, K
(direct method)
2I yFE FE
n
K x
or
dU
K E EG
da
K
Y
a
(x, y)
f)
a
g)
0
1
m
i i i
N
f i
i
i
a C K N
a a a
N N
Integration of Paris’ equationh)
af
ai
Number of cycles , N
Crackdepth,a
Fatigue Life
i)
Steps in Fatigue Life Prediction Procedure Based on the
da/dN- K Approach (cont’d)

No
j < L
Yes
CALCULATION of Tj
SAMPLING RANDOM VARIABLES
fa
aa
fC
CC
fS
SmaxS
STRUCTURAL COMPONENT
PWL
MATERIALLOADING
S
t
Failure probability calculation
Pf = P[T(X) Tr]:
L
)TrT(L
P f
f
fKth
KthKth
fK
KtK
n
K
da
Kth
dN
fKc
KIcKc
Probabilistic analysis using MC simulation
The FALPR statistical simulation flow chart for the analysis of fatigue crack growth

Irregular geometrical shape of a real fatigue crack

Welded Joint with Load = 4000 lb (ai/ci=0.286)
0
50000
100000
150000
200000
250000
300000
0 0.5 1 1.5 2 2.5
Crack Length 2c (in)
FatiguelifeNf(Cycle)
Exp. (Specimen #11)
Exp. (Specimen #12)
Exp. (Specimen #14)
Exp. (Specimen #15)
Exp. (Specimen #16)
Exp. (Specimen #17)
Exp. (Specimen #18)
Pre. Life (Loc. 1) No Rs
Pre. Life (Loc. 1) With Rs

Global and Local Approaches to Stress
Analysis and Fatigue

22
a
11
a
32
a=0
33
a
31
a 13
a= 0
12
a= 0
21
a=
0
23
a
11
22
33
0 0
0 0
0 0
a
a a
ij
a
Stress state near the notch tip (on the symmetry line)
1
2
3

22
a
11
a= 032
a
33
a
31
a 13
a= 0
12
a= 0
21
a=
0
23
a
11
22 23
32 33
0 0
0
0
a
a a a
ij
a a
Stress state in the disk at the blade-disk interface
1
2
3

1
peak
n
22
11
33
3
2
A D B
C
F
F
A, B
22
22
22
22
33
11
C
D
n
net
p t neak
F
A
and
K
Stresses concentration in axis-symmetric notched body

Stresses concentration in a prismatic notched body
net
pea
n
ntk
F
A
and
K
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

How to get the nominal stress from the Finite
Element Method stress data?
The smallest optimum element
size < ¼r !!
r

;
;
,
,
?
?
i
i
ak
n
p
i
e i
F F
F F
f f
g g
F
t
Fi
Fi+1
Fi-1
0
n
pea
k

Loads and Stresses
The load, the nominal stress, the local peak stress and the stress concentration factor
, ,
;
;
; ;
;
;
;
1
;
peak
tF F
Fpeak i i
F
Fn i i
n
net
n F
F
n
Fpe
et
ak
h h K
F
hk F F
k
F
A
k F
k
A
h F
Axial load – linear elastic analysis
n
y
dn
0
T
peak
Stress
F
F
Analytical, FEM Hndbk

Loads and Stresses
, ,
,
;
;
; ;
b
tMpeak i pe Mi ak i i
net
n
net
n M
net
M Mn i i
net
M or kh M K
M c
I
k M
c
k k M
I
Bending load – linear elastic analysis
n
y
dn
0
T
peak
Stress
M
b) M

peak eak
t t
n
p
oK r
S
K
Kt – stress concentration factor
(net or gross, net Kt gross Kt !! )
peak – stress at the notch tip
n - net nominal stress
S - gross nominal stress
Stress Concentration Factors in Fatigue Analysis
The nominal stress and the stress concentration factor in simple load/geometry configurations
n
net gross
P P
or S
A A
grossnet
n
net gross
M cM c
or S
I I
Simple axial load
Pure bending load
n
y
dn
0
T
r
peak
Stress
S
S
Tension
n
y
dn
0
T
r
peak
Stress
M
S
S
M
Bending
net Kt
gross Kt

Stress concentration factors for notched machine components
(B.J. Hamrock et. al., ref.(26)
d b
h
1.0
1.4
1.8
1.2
1.6
2.0
2.2
2.4
2.6
2.8
3.0
StressconcentrationfactorKt=peak/n
Diameter-to-width ratio d/b
0 0.1 0.2 0.3 0.4 0.5 0.6
peak
P P
n =P/[(b-d)h]

Stress concentration factors for notched machine components
(B.J. Hamrock et. al.
1.0
1.4
1.8
1.2
1.6
2.0
2.2
2.4
2.6
2.8
3.0
0 0.05 0.10 0.15 0.20 0.25 0.30
H/h=6
H/h=2
H/h=1.2
H/h=1.05
H/h=1.01
StressconcentrationfactorKt=peak/n
Radius-to-height ratio r/h
peak
r
M
h
b
M
H
2
6
n
Mc M
I bh

Various stress distributions in a T-butt weldment with transverse fillet welds;
r
t
t1
E
D
B
C
A
pea
k
n
hs
F
P
M
C
• Normal stress distribution in the weld throat plane (A),
• Through the thickness normal stress distribution in the weld toe plane (B),
• Through the thickness normal stress distribution away from the weld (C),
• Normal stress distribution along the surface of the plate (D),
• Normal stress distribution along the surface of the weld (E),
• Linearized normal stress distribution in the weld toe plane (F).
Stress concentration & stress distributions in weldments

0.65
1 exp 0.9
2 1
1 2
2.8 21 exp 0.45
2
ten
t
W
h h
K
W rW
th
: 2 0.6 pwhere W t h h
Range of application - reasonably designed weldments, (K.Iida and T. Uemura, ref. 14)
Stress concentration factor for a butt weldment
under axial loading
g=h
r
t
l = hp
PP

Stress concentration factor for a T-butt
weldment under tension load; (non-load carrying fillet weld)
0.65
1 exp 0.9
2 1
1
2.8 21 exp 0.45
2
t
t
W
h h
K
W rW
th
: 2 0.3 2p pwhere W t h t h
Validated for : 0.02 r/t 0.16 and
30o 60o
, source [14]
t
r
t1= tp
h
p
h
PP
y
x

Cyclic Loads and Cyclic Stress Patterns
(histories) in Engineering Objects

Loads and Stresses
,
,
, ;
;
;i
b
ti
t
F Mn i i
tpeak i
F Mpea
i
F M
k
i
ii
M
M
K
k F k
K
h
M
k F k
or
hF
Simultaneous axial and bending load
n
y
dn
0
T
peak
Stress
M
b) M
F
F
;
,
,
,
;
;
b
t
t
t
F M
F M
K
h
k
K
h
k From simple analytical stress analysis
From stress concentration handbooks
From detail FEM analysis

3
3
32
;
; ; ;
4 2 64
b b
b
eakn p
M c M
S
I d
W d d
M L l c I
Note! In the case of smooth components,
such as the railway axle, the nominal stress and the local peak stress are the same!
b)
d N.A.
,min 3
32
n
bM
d
,min 3
32
n
bM
d
1
2
3
n,max
n,min
time
Stress
n,a
1 cycle
c)
1
2
3
AB
y
x
L
l
Moment Mb
a)
RARB
W/2
W/2 W/2
W
The load W and the nominal stress n in an railway axle

22
A B C
t
t
23
0
0
A
B
C
23
2
20
A B C
t
t
23
0
0
22
A
B
C
23
22
0
Non-proportional loading path Proportional loading path
F
2R
t
x2
x3
F
T
T
22
33
22
33
x2
x3
23
Fluctuations and complexity of the stress state at the
notch tip

How to establish the nominal stress history?
a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e.
P=1, Mb =1, T=1 in order to establish the proportionality factors, kP, kM, and kT such that:
;;P M T
n n n
P M Tb
k P k M k T
b) The peak and valleys of the nominal stress history n,,i are determined by scaling the peak and
valleys load history Pi, Mb,I and Ti by appropriate proportionality factors kP, kM, and kT such that:
, , ,;
,
P M T
n i n i n i iiP M Tb i
k P k M k T
c) In the case of proportional loading the normal peak and valley stresses can be added and the
resultant nominal normal stress history can be established. Because all load modes in proportional
loading have the same number of simultaneous reversals the resultant history has also the same
number of resultant reversals as any of the single mode stress history.
;
,, i Mi Pn b i
k P k M
d) In the case of non-proportional loading the normal stress histories (and separately the shear
stresses) have to be added as time dependent processes. Because each individual stress history
has different number of reversals the number of reversals in the resultant stress history can be
established after the final superposition of all histories.
ii in P M b
t tt k P k M

Two proportional modes of loading
0
Mode a
Stressn,b
0
Stressn,a
0
Mode b
Stressn
Resultant stress: n= n,a+ n,b
0
Mode a
Stressn,b
0
Stressn,a
0
Mode b
Stressn
Resultant stress:
n(ti)= n,a(ti)+ n,b(ti)
time
time
Two non-proportional modes of loading
Superposition of nominal stress histories induced by two
independent loading modes

Wind load and stress fluctuations in a wind
turbine blade
Note! One reversal of the wind speed results in several stress reversals
Wind speed fluctuations + Blade vibrations Stress fluctuations
Source [43]
In-plane bending Out of plane bending
Time [s]
600 605 610 615 620
Time [s]
600 605 610 615 620
Windspeed[m/s]Load-lagstress[MPa]
- 5
5
0
8
10
12
14
- 40
0
10
12
14
- 50
- 30
- 28
Windspeed[m/s]Load-lagstress[MPa]

a) Ground loads on the wings, b) Distribution of the wing bending moment induced by the ground
load, c) Stress in the lower wing skin induced by the ground and flight loads
Characteristic load/stress history in the aircraft wing skin
time
Stress
0
Source [9]
0
a)
LandingTaxiing
Flying
b)
c)

;
;
,
,
?
?
i
i
ak
n
p
i
e i
F F
F F
f f
g g
F
t
Fi
Fi+1
Fi-1
0
n
pea
k

Loading and stress histories
Time
0
Stresspeak
Time0
LoadF
Time
Fi-1
Fi
Fi+1
Fmax
0
Stressn
n,i-1
n,i
n,i+1
n,max
Load history, F
Nominal stress history, n Notch tip stress history, peak
0
i-1
i
i+1
+1
-1
,n i
max
,
,
,
,
,ma
,
,max
,
,
x
1 1
F i
t t F i
i
peak i
peak i
p
n i
n i
n i
n
eakn
i
n i
k F
K K k F
F
F
Characteristic non-dimensional load/stress history
peak,
i
peak,i-1
peak,i+1
peak,max

How to get the nominal stress n from the
Finite Element method stress data?
Notched shaft under axial, bending and torsion load
a) Run each load case
separately for an unit
load
b) Linearize the FE stress
field for each load case
x3
F
r
D
t
x2F
T
T MM
d
Discrete cross section stress distribution
obtained from the FE analysis
d
22
0
n
x3
pea
k

0 0
1
3 2 2
1 6 6
2
;
1
12
n
i i
b t t
n
t
y y dy y y dy y y y
c M
tI t t
0
2
2
0
1 1
6
6
b t
n
n
m
n
i
t
n
ii
n
yy dy
t
y y
ydy
t
y y y
tt
0 0
1
1
;
1 1
n
i
m t t
n
y dy y dy y y
P
t t t t
Determination of
nominal stresses
from discrete FE data
by the linearization
method
x
y
yi
(yi)
yi
yn
(yn)
(y1)
yn
t1
peakn
t

How to get the resultant stress distribution from the
Finite Element stress data? (Notched shaft under axial, bending load)
x3
P
22
33
r
D
t
x
2
P
23
MM
d
d
22
0
n
b
x3
Bending
d
x3
22
0
n
m
Axial
d
220
pea
k
Resultant
x3
(x3)

Cyclic nominal stress and corresponding fluctuating stress distribution
Stressn
time
n, max
n, 0
n, min
x3
d
22
0
Resultant
22(x3, n,max)22(x3, n,0)22(x3, n,min)

• Multiaxial state of
stress at weld toe
• One shear and two
normal stresses
• Due to stress
concentration, xx is
the largest component
– Predominantly responsible
for fatigue damage
zz
xx
xx
zz
zx
xz
The stress state at the weld toe

Determination of the nominal, n, and the hot spot
stress , hs, from 3D-FE stress analysis data
a) Stress distribution in the critical cross section near the cover plate ending and the nominal or the
hot spot stress n (independent of length L ) and hs (independent of length L),
b) Stress distribution in the critical plane near the ending of a vertical attachment (gusset) and the
nominal or the hot spot stress n (dependent on length L ) or hs (independent of length L)
L
L t
m b t
hssh
n
s
t
h
x y dxdy
P
t L
x y d x y ydy
t
t
t
y
L
/2 0
/2
00
2
,
6 0,0,
- depends on L and is constant along the weld toe line
Independent of L but it changes
along the weld toe line
y
x
P
P
(x,y)
a)
L
t
peak
hs
y
x
PP
(x,y)
b)
L
t

The Nominal Stress n versus the local Hot Spot Stress hs
y
P
P
(x,y)
t
x
A
B
x
y
hs,B
hs,A
L
m b m b
hs A hsh A hs Bs A hs hs BB, ,, , , ,; ;
n A
n B
P
t L
,
,
;
??;

Example:
Preparation and Analysis of Representative
Stress/Load History:
The Rainflow Cycle Counting Procedure

Stress/Load Analysis - Cycle Counting Procedure
and Presentation of Results
The measured stress, strain, or load history is given usually in the form of a time series, i.e. a sequence of
discrete values of the quantity measured in equal time intervals. When plotted in the stress-time space the
discrete point values can be connected resulting in a continuously changing signal. However, the time effect
on the fatigue performance of metals (except aggressive environments) is negligible in most cases.
Therefore the excursions of the signal, represented by amplitudes or ranges, are the most important
quantities in fatigue analyses. Subsequently, the knowledge of the reversal point values, denoted with large
diamond symbols in the next Figure, is sufficient for fatigue life calculations. For that reason the intermediate
values between subsequent reversal points can be deleted before any further analysis of the loading/stress
signal is carried out. An example of a signal represented by the reversal points only is shown in slide no. 141.
The fatigue damage analysis requires decomposing the signal into elementary events called ‘cycles’.
Definition of a loading/stress cycle is easy and unique in the case of a constant amplitude signal as that one
shown in the figures. A stress/loading cycle, as marked with the thick line, is defined as an excursion starting
at one point and ending at the next subsequent point having exactly the same magnitude and the same sign
of the second derivative. The maximum, minimum, amplitude or range and mean stress values characterise
the cycle.
Unfortunately, the cycle definition is not simple in the case of a variable amplitude signal. The only non-
dubious quantity, which can easily be defined, is a reversal, example of which is marked with the thick line in
the Figures below.
max min
max min
max min
2
2
a
m s
stress range
stress amplitude
mean stres
max
min
time
Stress a
1 cycle
0
m

Removing material from a clay mine in Tennessee

Bending Moment Time Series
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
20
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81
Load point No.
Bendingmoment[10kNm]
Bending Moment measurements obtained at constant time intervals

Bending Moment History - Peaks and Valleys
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
20
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Load point No.
Bendingmomentvalue[10kNm]
Bending Moment signal represented by the reversal point values

Constant and Variable Amplitude Stress Histories;
Definition of the Stress Cycle & Stress Reversal
max min
max min
max min
min
max
;
2 2
;
2
a
m
R
Stress
Time
0
Variable amplitude stress history
One
reversal
b)
0
One cycle
mean
max
min
Stress
Time0
Constant amplitude stress historya)

Stress Reversals and Stress Cycles in a Variable
Amplitude Stress History
The reversal is simply an excursion between two-consecutive reversal
points, i.e. an excursion between subsequent peak and valley or valley
and peak.
In recent years the rainflow cycle counting method has been accepted
world-wide as the most appropriate for extracting stress/load cycles for
fatigue analyses. The rainflow cycle is defined as a stress excursion,
which when applied to a deformable material, will generate a closed
stress-strain hysteresis loop. It is believed that the surface area of the
stress-strain hysteresis loop represents the amount of damage induced
by given cycle. An example of a short stress history and its rainflow
counted cycles content is shown in the following Figure.

Stress
Time
Stress history Rainflow counted cycles
i-1
i-2
i+1
i
i+2
Stress History and the “Rainflow” Counted Cycles
1 1i i i iABS ABS
A rainflow counted cycle is identified when any two adjacent reversals in the
stress history satisfy the following relation:

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
2
i i
a
ABS
The stress range of such a cycle is:
1i iABS
The mean stress of such a cycle is:
1
2
i i
m
The Mathematics of the Cycle Rainflow Counting Method
for Fatigue Analysis of Fluctuating Stress/Load Histories

The rainflow cycle counting procedure - example
Determine stress ranges, Si, and corresponding mean stresses, Smi for the stress history
given below. Use the ‘rainflow’ counting procedure.
Si= 0, 4, 1, 3, 2, 6, -2, 5, 1, 4, 2, 3, -3, 1, -2 (units: MPa 102)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1
2
3
4
5
6
0
-1
-2
-3
StressSi(MPa102)
Reversing point number, i

The ASTM rainflow counting procedure
1. Find the reversing point with highest absolute stress magnitude,
2. The part of the stress history before the maximum absolute attach to the end of
the history,
3. Perform the rainflow counting on the re-arranged stress history, i.e. from
maximum to maximum
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5
Original stress history
Absolute maximum !

The ASTM modification of the Stress History
1 2 3 4 5 6 7 8 9 11 12 13 14 15
0
10
1
2
3
4
5
6
-1
-2
-3
The modified
stress history
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5
Absolute maximum
The original stress history

The Modified Stress History according to the ASTM
1 2 3 4 5 6 6 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
Stress(MPa)x102
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
starting point
Start counting from point No. 2 !

76
76 7 6 ,6 7
3 23 2 1; 2.5;
2 2m
54
54 5 4 ,4 5
1 41 4 3; 2.5;
2 2m
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
Start counting from the point No. 2 !!

2 3
2 3 2 3 ,6 7
2 52 (5) 7; 1.5;
2 2m
9 10
9 10 9 10 ,9 10
1 21 ( 2) 3; 0.5;
2 2m
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6

1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
13 14
13 14 13 14 ,13 14
3 23 2 1; 2.5;
2 2m
11 12
11 12 11 12 ,11 12
4 14 1 3; 2.5;
2 2m
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6

1 8
1 8 1 8 ,1 8
6 36 ( 3) 9; 1.5;
2 2m
1 2 3 4 5 6 7 8 9 11 12 13 14 15
-3
-2
-1
0
1
2
3
4
5
6
Cycles counted –ASTM method
1. 6- 7 =1; m,6- 7 = 2.5;
2. 4- 5 =3; m,4- 5 = 2.5;
3. 13- 14=1; m,10- 11= 2.5;
4. 11- 12=3; m,11- 12= 2.5;
5. 2- 3 =7; m,2- 3 = 1.5;
6. 9- 10 =3; m,9- 10 =-0.5;
7. 1- 8 =9; m,1- 8 = 1.5;

Extracted rainflow cycles, m
Total number of cycles, N=854
m -32 -22 -13 -3.2 6.44 16.1 25.7 35.3 45 54 64.1 73.7 83.3 92.9 103 112 122 131 141 151
298.8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
283.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
268.9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
254 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
239 0 0 0 0 0 0 0 1 2 2 0 0 0 0 0 0 0 0 0 0 5
224.1 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 5
209.2 0 0 0 0 0 0 0 3 4 5 2 0 0 0 0 0 0 0 0 0 14
194.2 0 0 0 0 0 1 0 1 7 2 0 0 0 0 0 0 0 0 0 0 11
179.3 0 0 0 0 0 0 1 0 4 4 0 0 0 0 0 0 0 0 0 0 9
164.3 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 5
149.4 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 4
134.5 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 0 0 6
119.5 0 1 1 0 0 0 0 0 0 0 3 1 5 1 2 0 0 0 0 0 14
104.6 0 0 1 2 1 0 0 0 0 2 4 3 7 3 2 1 2 1 0 0 29
89.64 0 1 2 3 7 2 0 0 0 1 2 8 10 7 5 6 2 1 0 0 57
74.7 1 1 3 4 3 5 0 1 2 2 10 18 23 20 17 11 4 1 0 0 126
59.76 2 1 5 7 4 1 4 5 1 2 11 20 34 31 31 28 9 7 1 1 205
44.82 1 6 9 7 9 7 10 3 3 8 15 37 49 64 62 41 16 11 2 1 361
29.88 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14.94 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
854
Mean stress, m
Stressrange,

Extracted rainflow cycles, m

b) the stress range
frequency distribution
diagram
Number cycles N
Stressrangej
j=7j=6
nj=6 - number of cycles j=6
in the class j=6
Total number of cycles in the entire history, NT
max
Relativestressrangej/max
Relative number of cycles N/NT
0
j=7j=6j=4321
nj=6/NT
max/ max1
j=4/ max
0.5 1.0
0.5
0
j=4321
a) The stress range
exceedance diagram
(stress spectrum)

Day 1
The End

Advances in fatigue and fracture mechanics by grzegorz (greg) glinka

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