Fatigue_Methods.pdf

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© 2014 Grzegorz Glinka. All rights reserved. 2

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, 3 rd 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 Netherl ands, 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 , Durabili ty and Stability of Welded Structures in Earth MovingMachinery” , Mashinos troenie, 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

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-Strain Analysis

Damage Analysis

Fatigue Life

MaterialProperties




Stress Parameters Used in Fatigue Analyses


K Y S a

a)

Sn

y

d n

0

T

epeak

S

y

d n

0

T

peak

M

b) c)

y

a

0

T

K a2

S

S

apeak

epeak

Sn crack

S n – net nominal stress; S – gross nominal stresse

peak – local linear-elastic notch-tip stress

apeak – local actual elastic-plastic notch-tip stress

Kt = epeak /S n – stress concentration factor

K – stress intensity factor



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

x

a0

T

S t r e s s ( x )

2 x

x

K

S

SThe Stress Intensity Factor K characterizingthe stress field in the crack tip region i sneeded!

The K factor can be obtained from :- ready made Handbook so lutions (easy to usebut 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 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 for Fatigue 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, K f

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

MATERIAL

No N

n

e ADF




a) Structu re

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.

K0K1K2K3K4K5

S t r e s s a m p l i t u d e ,

n / 2 o r

h s / 2

Number of cycl es, N0

The Simili tude 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

K0K1K2K3K4K5

S t r e s s a m p l i t u d e ,

n / 2 o r

h s / 2

Number of cycl es, N

d) Standard welded joints

c) Section with welded joint

a) Structure

b) Component

V

PR

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

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 1

2

3 4

5

S t r e s s

t

g)

Fatigue damage:h)

Total damage:i)

Fatigue life: N blck =1/D j)

S t r e s s

, 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

1

1

1;

m

D N A

31 52 4

1 1;

1 1 1 1 1 R R R f L

N L

N N

N n

D N N

2

22

1;

m

D N A

4

4

4

1;

m

D N A

5

10; D 42 51 3

5

5

1

32 41

1 1 1 1 1;

1 !!

i

i

D

N

D

N

D

N

D D

N N

D

if D Failure

e

N0N2

N1 N3

S t r e s s r a n g e

,

N4

N5

Cycles

Ni(

i)m = A or N

i=A/(

i)m

n R

3

33

1;

m

D N A



The FALSN fatigue life estimation sof tware – Typical input and output data

Weldment




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

S t r e s s a m p l i t u d e S

a [ N / m m

2 ]

Number of cycles N

400

450

500

10 5 10 6 10 7

(source: S. Nishijima, ref. 39)

S45 Steel temperedat 600 o C (W1)

Probability of failureP(%)

P=99%

P=90%

P=50%

P=10%

P=1%

8

6

48

10 8

f(N)

f(S)

m

a A N




N

j < LT

COMPUTING of T j

SAMPLING A SET OF RANDOM DATA

f K

Kf K

f R

RwR

f S

S maxS

STRUCTURAL COMPONENT

PWL

MATERIAL

No N

S

Rw

LOADING

S

t

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

( ) f f

L T Tr P

L

Probabilistic fatigue life assessment




Over designedUn-satisfactory Most

frequent

Characterist ic regions of cumulativeprobability of the fatigue life distribution




LOADING

F

t

GEOMETRY, K f

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

Information path for fatigue lifeestimation based on the -N method

MATERIAL

2N f




a) Specimen

b) Notched component

peak x

y

z

peak

peak

'

'

2 22

b f f f f N N c E

l o g ( / 2 )

lo g

(2N f )

f

0

l)

j)

The Similitude Concept states that if the local notch-tip strain history in thenotch tip and the strain history in thetest specimen are the same, then the

fatigue response in the notch tip regionand in the specimen will also be thesame and can be described by thematerial strain-life ( -N) curve.

The Simili tude Concept in the – N Method




Steps in fatigue life prediction procedure based onthe - N approach

a) Structure

b) Component

c) Section with welded joint

d)

peak

n

peak

hs

nhs




Fatigue damage:

1 2 3 41 2 3 4

1 1 1 1; ; ; ; D D D D

N N N N

Total d amage:

1 2 3 4 ; D D D D D

Fatigue life: N blck =1/D

e )

l o g ( / 2 )

'

'2 22

b c f f f f N N

E

e /E

log(2N f )

f /E

f

02N e

2N

2

: peak

Neuber E

' '

' '

'

'

,

,

,

?,?

f f

f f

f

f

f N

p e=

p e a k

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'

68

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 CrackInitiation 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: T i

i < LY

SAMPLING: k , K t , f , f ' , K’

N

COMPONENT

f(K t)

SCF K t

K5

S S

Computing of failure probabilities

( ) f r

f r

L T T P P T X T

L

MATERIAL

2N 1

K'

f M1

’f e

f M3

K’M

f M2

’f s

2 p

2, f

, f

E

2




P r o b a b i l i t y o f f a i l u r e




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

LOADING

F

t

GEOMETRY, K t

PSO

MATERIAL

0

E


Damage Analysis

Fatigue Life

MATERIAL

n

KKth

da

dN




The Similitude Concept states that i f 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 wi ll 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

C r a c k G r o w t h R a t e

, m / c y c l e

mMPa,K

The Similitude Concept in the da/dN – K Method






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)

2 I yFE FE

n

K x

or

dU K E EGda

K Y

a

(x, y)

f)

a

g)

0

1

m

i i i N

f ii

i

a C K N

a a a

N N

Integration of Paris’ equationh)

a f

a i

Number of cycles , N

C r a c k d e p t h

, a

Fatigue Life

i)

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













No

j < LYes

CALCULATION of T j

SAMPLING RANDOM VARIABLES

f a

aa

f C

CC

f S

S maxS

STRUCTURAL COMPONENT

PWL

MATERIALLOADING

S

t

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

L)Tr T(LP f

f

f Kth

KthKth

f K

KtK

n

K

da

Kth

dN

f Kc

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







Global and Local Approaches to Stress Analysis and Fatigue




22a

11a

32a =0

33a

31a 13

a = 0

12a = 0

21a =

0 23

a

11

22

33

0 0

0 0

0 0

a

a aij

a

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

1

2

3




22a

11a = 0 32

a

33a

31a 13

a = 0

12a = 0

21a =

0 23

a

11

22 23

32 33

0 0

0

0

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

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

n t k

F

A and

K

A, B, C

22

22

22

33

D

E

11

peak

n

22

11

33

3

2

AD B

C

F

F

1

E







;

;,

, ?

?

i

i

ak

n

p

i

e i F F

F F

f f

g g

F

t

F i

F i+1

F i-1

0

n pea

k




Loads and StressesThe load, the nominal stress, the local peak stress and the stress concentration factor

, ,

;

;

; ;

;

;

;

1

;

peak t F F

F peak i i

F

F n i i

n net

n F

F n

F pe

et

ak

h h K F

h k F F

k

F A

k F

k A

h F

Axial load – linear elas ti c analysis

n

y

d n

0

T

peak

S t r e s s

F

F

Analyt ical, FEM Hndbk




Loads and StressesThe load, the nominal stress, the local peak str ess and the stress concentration factor

, ,

,

;

;

; ;

b t M peak i pe M i ak i i

net n

net

n M

net M M n i i

net

M or k h M K

M c I

k M

c k k M I

Bending load – linear elastic analysis

n

y

d n

0

T

peak S t r e s s

M

b) M




peak eak t t

n

p o K rS

K

Kt – st ress concentrat ion factor (net or gross, net K t gross K t !! )

peak – st ress at th e notch t ip

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

n net gross

P or S

A A

gross net

n net gross

M c M c or S

I I

Simple axial load

Pure bending load

n

y

d n

0

T

r

peak

S t r e s s

S

S

Tension

n

y

d n

0

T

r

peak S t r e s s

MS

SM

Bending

net K tgross K t






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=2

H/h=1.2H/h=1.05H/h=1.01

S t r e s s c o n c e n t r a t i o n f a c t o r K

t = p e a k / n

Radius-to-height ratio r/h

peak

r

Mh

b

MH

2

6 n

c M I 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 1

1 22.8 21 exp 0.45

2

ten t

W h h

K W rW t h

: 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 = h p

PP

St t ti f t f T b tt




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

0.65

1 exp 0.9 2 11

2.8 21 exp 0.452

t t

W h h

K W rW t h

: 2 0.3 2 p pwhere W t h t h

Validated for : 0.02 r/t 0.16 and30 o 60 o

, source [14]

t

r

t1= t p hp

h

PP

y

x



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






3

3

32 ;

; ; ;4 2 64

b b

b

eak n 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 str ess and the local peak stress are the same!

b)

d N.A.

,min 3

32 n

b M d

,min 3

32 n

b M d

1

2

3

n,max

n,min

time S t r e s s

n,a

1 cyclec)

1

2

3

AB

y

x

L

Moment M b

a)

R ARB

W/2W/2 W/2

W

The load W and the nominal stress n in an railway axle


Fluctuations and complexity of the stress state at the



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

2 R

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, M b =1, T=1 in order to establish the proportionality factors, k P, k M, and k T such that:

;; P M T n n n

P M T b k 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 P i, Mb,I and Ti by appropriate proportionality factors k P, k M, and k T such that:

, , ,;, P M T n i n i n i ii P M T b i 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 M i P n b i 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 historyhas different number of reversals the number of reversals in the resultant stress history can beestablished after the final superposition of all histories.

ii i n M b t t t k P k M


Superposition of nominal stress histories induced by two



Two p roportional modes of lo ading

0

Mode a

S t r e s s

n , b

0

S t r e s s

n , a

0

Mode b

S t r e s s n

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

0

Mode a

S t r e s s

n , b

0

S t r e s s

n , a

0

Mode b

S t r e s s n

Resultant stress:n(t i)= n,a (t i)+ n,b (t i)

time

time

Two non-proportional modes of loading

Superposition of nominal stress histories induced by twoindependent loading modes




Wind load and stress f luctuations 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

W i n d s p e e d [ m / s ]

L o a d - l a g s t r e s s [ M P a ]

- 5

5

0

8

10

12

14

- 400

10

12

14

- 50

- 30

- 28

W i n d s p e e d [ m / s ]

L o a d - l a g s t r e s s [ M P a ]




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

S t r e s s

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

F i

F i+1

F i-1

0

n pea

k




How to get the nominal stress from the



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 case

separately for an unitload

b) Linearize the FE stressfield for each load case

x3

F

r

D

t

x2F

T

T MM

d

Discrete cross section stress distributionobtained fro m the FE analysis

d 22

0

nx3

peak


00




0 0

13 2 2

1 6 62

;1

12

n

i i b t t n

t y y dy y y dy y y y

c M t I t t

0

2

2

0

1 1

6

6

b t n

n

m

n

i

t n

ii

n

y y dy

t

y y

ydy

t

y y y

t t

0 0

1

1

;1 1

ni

m t t n

y dy y dy y y P

t t t t

Determination of nominal stressesfrom discrete FE databy the linearizationmethod

x

y

y i

(y i)

y iyn

(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

d d 22

0

nb

x3

Bending

d

x3

22

0

nm

Axial

d 220

peak

Resultant

x3

(x 3)


Cyclic nominal stress and corresponding fluctuating stress dist ribution



Cyclic nominal stress and corresponding fluctuating s tress dist ribution

S t r e s s

n

time

n, max

n, 0

n, min

x3

d22

0

Resultant

22(x 3, n,max )22(x 3, n,0 )22(x 3, n,min )


Th h ld




• Multiaxial state of stress at weld toe

• One shear and twonormal stresses

• Due to stressconcentration, xx isthe 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



n pstress , 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 t hs s h

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 changesalong the weld toe line

y

x

P

P

(x,y)

a)

L

tpeak

hs

y

x

PP(x,y)

b)

L

t


The Nominal Stress ers s the local Hot Spot Stress



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 b hs A hs h A hs B s A hs hs B B, ,, , , ,; ;

n A

n B

P t L,

,

;

??;




Example:Preparation and Analysis of Representative

Stress/Load History:

The Rainflow Cycle Counting Procedure









B di M Ti S i



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.

B e n d i n g m o m e n t [ 1 0

k N m ]

Bending Moment measurements obtained at constant time intervals




Bending Moment Histo ry - 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.

B e n d i n g m o m e n t v a l u e [ 1 0 k N m ]

Bending Moment signal represented by the reversal point values


Constant and Variable Amplitude Stress Histories;



Constant and Variable Amplitude Stress Histories;Definiti on of the Stress Cycle & Stress Reversal

max min

max min

max min

min

max

;

2 2

;2

a

m

R

S t r e s s

Time

0

Variable amplitude stress history

Onereversal

b)

0

One cycle

mean

max

min

S t r e s s

Time0

Constant amplitude stress historya)




Stress Reversals and Stress Cycles in a Variable Amplitude Stress History

The reversal is s imply 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 for fatigue 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.


Stress History and the “ Rainflow ” Counted Cycles



S t r e s s

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 i ABS ABS

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


The Mathematics of the Cycle Rainflow Counting Method



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

1 1i i i i ABS ABS

The stress amplitude of such a cycle is:

1

2i i

a

ABS

The stress range of such a cycle is:

1i i ABS

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



The rainflow cycle counting procedure - example

Determine stress ranges, S i, and corr esponding mean str esses, S mi for the stress historygiven below. Use the ‘ rainflow ’ counting procedure.

S i= 0, 4, 1, 3, 2, 6, -2, 5, 1, 4, 2, 3, -3, 1, -2 (units: MPa 10 2)

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

S t r e s s S

i ( M P a 1 0 2 )

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 hi story,

3. Perform the rainflow counting on the re-arranged stress histo ry, 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

1

23

45

6

-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 s tress history




Start counting from the point No. 2 !!



7676 7 6 ,6 7

3 23 2 1; 2.5;2 2

m

5454 5 4 ,4 5

1 41 4 3; 2.5;2 2 m

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




2 32 3 2 3 ,6 7

2 52 (5) 7; 1.5;2 2

m

9 109 10 9 10 ,9 10

1 21 ( 2) 3; 0.5;2 2 m

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 2

m

11 1211 12 11 12 ,11 12

4 14 1 3; 2.5;2 2 m

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 2

m

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 9164.3 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

S t r e s s r a n g e ,


Extracted rainflow cycles, m




Total number of cycles in th e entire history, NT

a) The stress range



b) the stress rangefrequency distributiondiagram

Number cycles N

S t r e s s r a n g e

j

j=7 j=6

n j=6 - number of cycles j=6

in the class j=6

max

R e l a t i v e s t r e s s r a n g e

j / m a x

Relative number of cycles N/N T

0

j=7 j=6 j=4321

n j=6 /NT

max / max1

j=4 / max

0.5 1.0

0.5

0

j=4321

a) The stress rangeexceedance diagram(stress spectrum)




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