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

$Page 1: Advances in fatigue and fracture mechanics by grzegorz (greg) glinka$
© 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, JohnWiley, 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 (inFrench 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 MovingMachinery”, 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 ofWelded 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

http://www.vintagecalculators.com/html/texas_instruments_ti-30.html


DAY 1

Contemporary Fatigue AnalysisMethods

(basics concepts and assumptions)

Information Path for Strength and Fatigue Life Analysis

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Fatigue Life

MaterialProperties


Stress Parameters Used in Fatigue Analyses


K YS a

a)

Sn

y

dn

0

T

epeak

S

y

dn

0

T

peak

M

b) c)

y

a0

T

Ka2

S

S

apeak

epeak

Sn crack

Sn – net nominal stress; S – gross nominal stresse

peak – local linear-elastic notch-tip stressa

peak – local actual elastic-plastic notch-tip stress

Kt = epeak /Sn – stress concentration factor

K – stress intensity factor

What stress parameter is needed for the FractureMechanics based (da/dN- K) fatigue analysis?

x

a0

T

Stre

ss(x

) 2x

xK

S

S The Stress Intensity Factor K characterizingthe stress field in the crack tip region isneeded!

The K factor can be obtained from:- ready made Handbook solutions (easy to usebut often inadequate in practice)

- from the near crack tip stress (x)distribution or the displacement data obtainedfrom FE analysis of a cracked body (tedious)

- from the weight function by using the FEstress 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 forFatigue Life Analysis - outlines

• Stress-Life Method or the S - N approach;uses the nominal or simple engineering stress ‘ S ‘ toquantify fatigue damage

• Strain-Life Method or the - N approach;uses the local notch tip strains and stresses to quantifythe fatigue damage

• Fracture Mechanics or the da/dN - K approach;uses the stress intensity factor to quatify the fatiguecrack growth rate


Information path for fatigue life estimation based onthe S-N method

LOADING

F

t

GEOMETRY, Kf

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

MATERIAL

No N

n

eADF


a) Structure

Q

H

F

K 5

n

The Similitude Concept states that if the nominal stress histories in the structure and in the testspecimen are the same, then the fatigue response in each case will also be the same and can bedescribed by the generic S-N curve. It is assumed that such an approach accounts also for the stressconcentration, loading sequence effects, manufacturing etc.

K0K1K2K3K4K5St

ress

ampl

itude

,n/2

orhs

/2Number of cycles, N

0

The Similitude Concept in the S-N Method


Steps in Fatigue Life Prediction Procedure Based on theS-N Approach

The S – N method

5

e) Standard S-N curves

K0K1K2K3K4K5St

ress

ampl

itude

,n/2

orhs

/2

Number of cycles, N

d) Standard welded joints

c) Section with welded joint

a) Structure

b) Component

VP

R

Q

H

F

Weld


5

5

5

5

5

11

1 5

22

2 5

33

3 5

44

4 5

55

5 5

1

1

1

1

1

m

m

m

m

m

DN C

DN C

DN C

DN C

DN C

1 2 3 4 5 ;D D D D D D1

2

3 45

Stre

ss

t

g)

Fatigue damage:h)

Total damage:i)

Fatigue life: N blck=1/Dj)

Stre

ss,

nf)

t

K5

n

Steps in Fatigue Life Prediction Procedure Basedon the S-N Approach (continued)



The linear hypothesis of Fatigue Damage Accumulation(the Miner rule)

1

11

1 ;m

DN A

31 52 4

1 1 ;1 1 1 1 1R RRfL

NL

NN

Nn

D NN

2

22

1 ;m

DN A

4

44

1 ;m

DN A

51 0;D 42 51 3

5

51

32 41

1 1 1 1 1 ;

1 !!

ii

D

N

D D

N

D

N

D D

NN

D

if D Failure

e

N0N2N1 N3

Stre

ssra

nge,

N4

N5

Cycles

Ni( i)m = A or Ni=A/( i)m

nR

3

33

1 ;m

DN A

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

Weldment


The scatter in fatigue:Fatigue S-N curves for assigned probabilityof failure; P-S-N curves

Stre

ssam

plitu

deS a

[N/m

m2 ]

Number of cycles N

400

450

500

105 106 107

(source: S. Nishijima, ref. 39)

S45 Steel temperedat 600o C (W1)

Probability of failureP(%)

P=99%

P=90%

P=50%

P=10%

P=1%

8

648

108

f(N)

f(S)

ma A N


N

j < LT

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 calculationPf = P[T(X) Tr]:

( )ff

L T TrP

L

Probabilistic fatigue life assessment


Over designedUn-satisfactory Mostfrequent

Characteristic regions of cumulativeprobability of the fatigue life distribution


LOADING

F

t

GEOMETRY, Kf

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

Information path for fatigue lifeestimation based on the -N method

MATERIAL

2Nf


a) Specimen

b) Notched component

peakx

y

z

peak

peak

''2 2

2bf

f f fN N cE

log

(/2

)

log(2Nf)

f

0

l)

j)

The Similitude Concept states that ifthe local notch-tip strain history in thenotch tip and the strain history in thetest specimen are the same, then thefatigue response in the notch tip regionand in the specimen will also be thesame and can be described by thematerial strain-life ( -N) curve.

The Similitude Concept in the – N Method


Steps in fatigue life prediction procedure based onthe - N approach

a) Structure

b) Component


d)

peak

n

peak

hs

nhs


Fatigue damage:

1 2 3 41 2 3 4

1 1 1 1; ; ; ;D D D DN 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

2b cf

f f fN NE

e/E

log(2Nf)

f/E

f

02Ne2N

2

: peakNeuberE

' '

' '

'

'

,

,

,

?,?

f f

f f

f

f

fN

p e=

peak

t

1

2

3

4

5

6

7

8

1'

f)

0t

'1

'

n

E K

0

3

2,2'4

5,5'7,7'

68

1,1'

(continued). Steps in fatigue life prediction procedure based on the -N approach© 2008 Grzegorz Glinka. All rights reserved. 22




Three basic sets of input data for the evaluation of the Fatigue CrackInitiation Life and Reliability (the – N approach)

T T

R(T) = 1 - F(T)1f(T)

LOADINGS

t

f(k)

Scaling factor k

LIFE CALCULATION: Ti

i < LY

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 TP P T X T

L

MATERIAL

2N 1

K'

fM1

’fe

fM3

K’M

fM2

’fs

2p

2,f

,f

E

2


Prob

abili

tyof

failu

re


Information path for fatigue life estimation based onthe da/dN- K method

LOADING

F

t

GEOMETRY, Kt

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

MATERIAL

n

KKth

dadN


The Similitude Concept states that if the stressintensity K for a crack in the actual componentand in the test specimen are the same, then thefatigue crack growth response in the componentand in the specimen will also be the same andcan be described by the material fatigue crackgrowth curve da/dN - K.

a) Structure

Q

H

F

ab) Weld detail

c) Specimena

P

P

K 10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ckG

row

thR

ate,

m/c

ycle

mMPa,K

The Similitude Concept in the da/dN – K Method



VP

Q

R

H

F

Weld

a) Structure

b) Component


S 1

S 2

S3 S4

S 5

Stre

ss,S

t

e)

d)

Steps in the Fatigue Life Prediction Procedure Basedon 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

KYa

Stress intensity factor, K(direct method)

2I yFE FE

n

K xor

dUK E EGda

KYa

(x, y)

f)

a

g)

01

mi i i

N

f ii

i

a C K N

a a a

N N

Integration of Paris’ equationh)

af

ai

Number of cycles , N

Cra

ckde

pth,

a

Fatigue Life

i)

Steps in Fatigue Life Prediction Procedure Based on theda/dN- K Approach (cont’d)





No

j < LYes

CALCULATION of Tj

SAMPLING RANDOM VARIABLES

fa

aa

fC

CC

fS

SmaxS

STRUCTURAL COMPONENT

PWL

MATERIALLOADING

S

t

Failure probability calculationPf = P[T(X) Tr]:

L)TrT(LP 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© 2008 Grzegorz Glinka. All rights reserved. 35

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)

Fatig

uelif

eN

f(C

ycle

)

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 RsPre. Life (Loc. 1) With Rs



Global and Local Approaches to StressAnalysis and Fatigue


22a

11a

32a=0

33a

31a 13

a= 0

12a= 0

21a=

023

a

11

22

33

0 00 00 0

a

a aij

a

Stress state near the notch tip (on the symmetry line)

1

2

3


22a

11a= 032

a

33a

31a 13

a= 0

12a= 0

21a=

023

a

11

22 23

32 33

0 000

a

a a aij

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

nnet

p t neak

FA

andK

Stresses concentration in axis-symmetric notched body


Stresses concentration in a prismatic notched body

net

pea

n

ntk

FA

andK

A, B, C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

A D B

C

F

F

1E


How to get the nominal stress from the FiniteElement Method stress data?

The smallest optimum elementsize < 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 StressesThe load, the nominal stress, the local peak stress and the stress concentration factor

, ,

;

;

; ;

;

;

;

1

;

peaktF F

Fpeak i i

F

Fn i i

nnet

n F

Fn

Fpe

et

ak

h h KF

hk F F

k

FA

k F

k A

h F

Axial load – linear elastic analysis

n

y

dn

0

T

peak

Stre

ss

F

F

Analytical, FEM Hndbk



, ,

,

;

;

; ;

btMpeak i pe Mi ak i i

netn

net

n M

netM Mn i i

net

M or kh M K

M cI

k M

ck k MI

Bending load – linear elastic analysis

n

y

dn

0

T

peak

Stre

ssM

b) M


peak eakt t

n

poK rS

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 AnalysisThe nominal stress and the stress concentration factor in simple load/geometry configurations

nnet gross

P Por SA A

grossnetn

net gross

M cM cor S

I I

Simple axial load

Pure bending load

n

y

dn

0

T

r

peak

Stre

ss

S

S

Tension

n

y

dn

0

T

r

peak

Stre

ss

MS

SM

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

3.0St

ress

conc

entr

atio

nfa

ctor

Kt=

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=6H/h=2H/h=1.2H/h=1.05H/h=1.01

Stre

ssco

ncen

trat

ion

fact

orK

t=pe

ak/

n

Radius-to-height ratio r/h

peak

r

Mh

b

MH

26

nMc MI bh


Various stress distributions in a T-butt weldment with transverse fillet welds;

r

t

t1

ED

BC

A

pea

k

n

hs

FP

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 0.92 11 2

2.8 21 exp 0.452

tent

Wh hK

W rWth

: 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 weldmentunder axial loading

g=

h

r

t

l = hp

PP


Stress concentration factor for a T-buttweldment under tension load; (non-load carrying fillet weld)

0.65

1 exp 0.92 11

2.8 21 exp 0.452

tt

Wh hK

W rWth

: 2 0.3 2p pwhere W t h t h

Validated for : 0.02 r/t 0.16 and30o 60o

, source [14]

t

r

t1= tp hp

h

PP

y

x

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



,

,

, ;

;

;i

bti

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 kor

hF

Simultaneous axial and bending load

n

y

dn

0

T

peak

Stre

ss

M

b) M

F

F

;

,

,

,

;

;

bt

tt

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 pM c M

SI d

W d dM L l c INote! 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

32n

bMd

,min 3

32n

bMd

1

2

3

n,max

n,min

timeStre

ss n,a

1 cyclec)

1

2

3

AB

y

x

Ll

Moment Mb

a)

RARB

W/2W/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

F2R

t

x2

x3

F

T

T

22

33

22

33

x2

x3

23

Fluctuations and complexity of the stress state at thenotch 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 Tn n nP M Tbk P k M k T

b) The peak and valleys of the nominal stress history n,,i are determined by scaling the peak andvalleys load history Pi, Mb,I and Ti by appropriate proportionality factors kP, kM, and kT such that:

, , ,;,P M Tn i n i n i iiP M Tb ik P k M k T

c) In the case of proportional loading the normal peak and valley stresses can be added and theresultant nominal normal stress history can be established. Because all load modes in proportionalloading have the same number of simultaneous reversals the resultant history has also the samenumber of resultant reversals as any of the single mode stress history.

;,, i Mi Pn b ik P k M

d) In the case of non-proportional loading the normal stress histories (and separately the shearstresses) have to be added as time dependent processes. Because each individual stress historyhas different number of reversals the number of reversals in the resultant stress history can beestablished after the final superposition of all histories.

ii in P M bt tt k P k M


Two proportional modes of loading

0

Mode a

Stre

ssn,

b

0

Stre

ssn,

a

0

Mode b

Stre

ssn

Resultant stress: n= n,a+ n,b

0

Mode a

Stre

ssn,

b

0

Stre

ssn,

a

0

Mode b

Stre

ssn

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 twoindependent loading modes


Wind load and stress fluctuations in a windturbine 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

Win

dsp

eed

[m/s

]Lo

ad-la

gst

ress

[MPa

]

- 5

5

0

8

10

12

14

- 400

10

12

14

- 50

- 30

- 28

Win

dsp

eed

[m/s

]Lo

ad-la

gst

ress

[MPa

]


a) Ground loads on the wings, b) Distribution of the wing bending moment induced by the groundload, c) Stress in the lower wing skin induced by the ground and flight loads

Characteristic load/stress history in the aircraft wing skin

time

Stre

ss

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

Stre

sspe

ak

Time0

Load

F

Time

Fi-1

Fi

Fi+1

Fmax

0

Stre

ssn

n,i-1

n,i

n,i+1

n,max

Load history, F Nominal stress history, n Notch tip stress history, peak

0i-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 in

eakni

n i

k FK K k F

FF

Characteristic non-dimensional load/stress history

peak,

i

peak,i-1

peak,i+1

peak,max


How to get the nominal stress n from theFinite Element method stress data?

Notched shaft under axial, bending and torsion load

a) Run each load caseseparately for an unitload

b) Linearize the FE stressfield for each load case

x3

F

r

D

t

x2F

T

T MMd

Discrete cross section stress distributionobtained from the FE analysis

d 22

0

nx3

peak



0 0

13 2 2

1 6 62;

112

n

i ib t tn

t y y dy y y dy y y yc M

tI t t

0

2

2

0

1 1

6

6

b tn

n

m

n

i

tn

ii

n

yy dy

t

y y

ydy

t

y y y

tt

0 0

1

1;

1 1

n

im t tn

y dy y dy y yP

t t t t

Determination ofnominal stressesfrom discrete FE databy the linearizationmethod

x

y

yi

(yi)

yiyn

(yn)

(y1)

yn

t1

peakn

t

How to get the resultant stress distribution from theFinite Element stress data? (Notched shaft under axial, bending load)

x3

P22

33

r

D

t

x2

P 23

MM

dd 22

0

nbx3

Bending

d

x3

22

0

nm

Axiald 220

peak

Resultant

x3

(x3)


Cyclic nominal stress and corresponding fluctuating stress distribution

Stre

ssn

time

n, max

n, 0

n, min

x3

d

220

Resultant

22(x3, n,max)22(x3, n,0)22(x3, n,min)



• Multiaxial state ofstress at weld toe

• One shear and twonormal stresses

• Due to stressconcentration, xx isthe largest component– Predominantly responsible

for fatigue damage

zz

xx

xx

zz

zxxz

The stress state at the weld toe

Determination of the nominal, n, and the hot spotstress , hs, from 3D-FE stress analysis data

a) Stress distribution in the critical cross section near the cover plate ending and the nominal or thehot 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 thenominal or the hot spot stress n (dependent on length L ) or hs (independent of length L)

L

L t

m b thssh

n

st

h

x y dxdyP

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 changesalong the weld toe line

y

x

PP

(x,y)

a)

L

tpeak

hs

y

x

PP(x,y)

b)

L

t


The Nominal Stress n versus the local Hot Spot Stress hs

y

PP

(x,y)

t

x

A

B

xy

hs,B

hs,A

L

m b m bhs A hsh A hs Bs A hs hs BB, ,, , , ,; ;

n A

n B

Pt L,

,

;

??;


Example:Preparation and Analysis of Representative

Stress/Load History:

The Rainflow Cycle Counting Procedure


Stress/Load Analysis - Cycle Counting Procedureand Presentation of Results

The measured stress, strain, or load history is given usually in the form of a time series, i.e. a sequence ofdiscrete values of the quantity measured in equal time intervals. When plotted in the stress-time space thediscrete point values can be connected resulting in a continuously changing signal. However, the time effecton 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 importantquantities in fatigue analyses. Subsequently, the knowledge of the reversal point values, denoted with largediamond symbols in the next Figure, is sufficient for fatigue life calculations. For that reason the intermediatevalues between subsequent reversal points can be deleted before any further analysis of the loading/stresssignal 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 oneshown in the figures. A stress/loading cycle, as marked with the thick line, is defined as an excursion startingat one point and ending at the next subsequent point having exactly the same magnitude and the same signof the second derivative. The maximum, minimum, amplitude or range and mean stress values characterisethe 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 inthe Figures below.

max min

max min

max min

2

2

a

m s

stress range

stress amplitude

mean stres

max

min

timeStre

ssa

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.

Ben

ding

mom

ent

[10

kNm

]

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.

Ben

ding

mom

entv

alue

[10

kNm

]

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

RStre

ss

Time0

Variable amplitude stress history

Onereversal

b)

0

One cycle

mean

max

min

Stre

ss

Time0

Constant amplitude stress historya)


Stress Reversals and Stress Cycles in a VariableAmplitude Stress History

The reversal is simply an excursion between two-consecutive reversalpoints, i.e. an excursion between subsequent peak and valley or valleyand peak.

In recent years the rainflow cycle counting method has been acceptedworld-wide as the most appropriate for extracting stress/load cycles forfatigue analyses. The rainflow cycle is defined as a stress excursion,which when applied to a deformable material, will generate a closedstress-strain hysteresis loop. It is believed that the surface area of thestress-strain hysteresis loop represents the amount of damage inducedby given cycle. An example of a short stress history and its rainflowcounted cycles content is shown in the following Figure.


Stre

ss

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 thestress history satisfy the following relation:


A rainflow counted cycle is identified when any two adjacent reversals inthee stress history satisfy the following relation:

1 1i i i iABS ABSThe 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 Methodfor 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 historygiven 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

Stre

ssS i

(MPa

102 )

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 ofthe history,

3. Perform the rainflow counting on the re-arranged stress history, i.e. frommaximum 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 14 150

10

123456

-1-2-3

The modifiedstress 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

Stre

ss(M

Pa)x

102

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 !


7676 7 6 ,6 7

3 23 2 1; 2.5;2 2m

5454 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 32 3 2 3 ,6 7

2 52 (5) 7; 1.5;2 2m

9 109 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 1413 14 13 14 ,13 14

3 23 2 1; 2.5;2 2m

11 1211 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 81 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 method1. 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 151298.8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1283.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0268.9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1254 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1239 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 5209.2 0 0 0 0 0 0 0 3 4 5 2 0 0 0 0 0 0 0 0 0 14194.2 0 0 0 0 0 1 0 1 7 2 0 0 0 0 0 0 0 0 0 0 11179.3 0 0 0 0 0 0 1 0 4 4 0 0 0 0 0 0 0 0 0 0 9164.3 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 5149.4 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 4134.5 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 0 0 6119.5 0 1 1 0 0 0 0 0 0 0 3 1 5 1 2 0 0 0 0 0 14104.6 0 0 1 2 1 0 0 0 0 2 4 3 7 3 2 1 2 1 0 0 2989.64 0 1 2 3 7 2 0 0 0 1 2 8 10 7 5 6 2 1 0 0 5774.7 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 20544.82 1 6 9 7 9 7 10 3 3 8 15 37 49 64 62 41 16 11 2 1 36129.88 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 014.94 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

Stre

ssra

nge,


Extracted rainflow cycles, m


b) the stress rangefrequency distributiondiagram

Number cycles N

Stre

ssra

nge

jj=7j=6

nj=6 - number of cycles j=6in the class j=6

Total number of cycles in the entire history, NT

max

Rel

ativ

est

ress

rang

ej/

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 rangeexceedance diagram(stress spectrum)


Day 1

The End
