A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under...

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Fatigue Workshop Fatigue Workshop - - Broadband spectral fatigue: from Gaussian to non Broadband spectral fatigue: from Gaussian to non - - Gaussian, from Gaussian, from research to industry” research to industry” A comprehensive approach to fatigue A comprehensive approach to fatigue under random loading: under random loading: non non - - Gaussian and non Gaussian and non - - stationary loading investigations stationary loading investigations ENDIF ENDIF Dipartimento di Ingegneria Dipartimento di Ingegneria Università di Ferrara, Italy Università di Ferrara, Italy DIEGM DIEGM Dip. Dip. Ing Ing . . Elettrica Elettrica Gest Gest . . Meccanica Meccanica Università di Udine, Italy Università di Udine, Italy Denis Benasciutti Denis Benasciutti Roberto Roberto Tovo Tovo February 24 February 24 th th , 2010 , 2010 Paris (F) Paris (F)

Transcript of A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under...

Page 1: A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under random loading: ... rainflow count : ... Transformation of rainflow cycles A non-Gaussian

Fatigue Workshop Fatigue Workshop -- ““Broadband spectral fatigue: from Gaussian to nonBroadband spectral fatigue: from Gaussian to non--Gaussian, from Gaussian, from research to industry”research to industry”

A comprehensive approach to fatigueA comprehensive approach to fatigueunder random loading:under random loading:

nonnon--Gaussian and nonGaussian and non--stationary loading investigationsstationary loading investigations

ENDIFENDIFDipartimento di IngegneriaDipartimento di IngegneriaUniversità di Ferrara, ItalyUniversità di Ferrara, Italy

DIEGMDIEGMDip. Dip. IngIng. . ElettricaElettrica GestGest. . MeccanicaMeccanica

Università di Udine, ItalyUniversità di Udine, Italy

Denis BenasciuttiDenis Benasciutti Roberto Roberto TovoTovo

February 24February 24thth, 2010 , 2010 –– Paris (F)Paris (F)

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OverviewReal service loading :

•• RandomRandom•• nonnon--GaussianGaussian•• nonnon--stationarystationary•• multimulti--axialaxial

Planned research activity steps

1. Stationary, Gaussianuniaxial loading

2. non-Gaussian loading

3. non-stationary loading

4. multi-axial loading

Int J Fatigue (2002, 2005)Prob Eng Mechanics (2006)

Prob Eng Mechanics (2005)Int J Fatigue (2006)

This presentation:

• Introduction & theoretical background

• Gaussian loadings

• non-Gaussian loadings. Case study: mountain-bike data, automotive application

• non-stationary loadings (only a brief introduction)

This presentation:

• Introduction & theoretical background

• Gaussian loadings

• non-Gaussian loadings. Case study: mountain-bike data, automotive application

• non-stationary loadings (only a brief introduction)

Fat Fract Eng Mat Struct (2007)"VAL 2" Conference (2009)

Int J Mat & Product Tech (2007)Fat Fract Eng Mat Struct (2009)

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Fatigue analysis of random loadings

FATIGUE LIFE

??

TIME DOMAINTIME DOMAIN FREQUENCY DOMAINFREQUENCY DOMAIN

DAMAGE – FATIGUE LIFE

Force \ stress \ strain

Time

amplitude

n° cycles

CYCLE DISTRIBUTIONamplitude

n° cumulated cycles (log)

LOADING SPECTRUM

COUNTING METHOD(e.g. ‘rainflow’ counting)

DAMAGE ACCUMULATION RULE(e.g. Palmgren-Miner linear law)

Frequency

PSD

Force \ stress \ strain

Time

•• randomrandom•• uniaxialuniaxial•• stationarystationary

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Stationary random loadingsStationary random loadings

GAUSSIAN• narrow-band

• broad-band

- Rayleigh amplitude PDF

- Wirsching & Light (1980)- Dirlik (1985)- Zhao & Baker (1992)- Tovo (2002), Benasciutti & Tovo (2005)- Markov approach (Rychlik)

NON-GAUSSIAN• narrow-band

- Hermite model (Winterstein 1988)- power-law model (Sarkani et al. 1994)

• broad-band- Yu et al. (2004)- Benasciutti & Tovo (2005)- Markov approach- trasformed model (Rychlik)

STATIONARY LOADING

time

s(t)

ω

G(ω)

Gaussian

non-Gaussian

40

22

20

11 ;

0

ii dωωGωλ

Spectral parameters :

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-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8

Max u

min

v

Fatigue analysis of random loadings

For repeated measurements (in the same condition):

{C1 , C2 , ... , Cn1}

{C1 , C2 , ... , ... , Cn2}

{C1 , ... , Cnk}

Counted cycle: (u,v)

MEASURED LOADMEASURED LOAD

+ +

RAINFLOWRAINFLOW CYCLES CCYCLES Cii

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-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8

minv

Max u

Is olines of h(u,v) joint PDF

s

m

u v

dydxy)h(x,v)H(u,

Cycle distribution in random loadingsCycle distribution in random loadings

v)h(u, joint PDF

CDF

u

v v][u,dv duv)h(u, Prob

u = s + m

v = s - m

ms

s

m)sm,h(s2m)(s,p ma,

-

ma,a dmm)(s,p(s)p

amp-mean PDF

amp. PDF

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Loading spectrum and fatigue damageLoading spectrum and fatigue damage

s

a dx(x)p(s)F

fatigue loading spectrum

0

amm ds(s)pss

Ks(T)N(T)D

m

fatigue damage

(s)F

s

PDF , CDF

h(u,v) , H(u,v)

amp. PDF

pa(s)

damage

D

LES

S “I

NFO

RM

ATI

ON

”LE

SS

“IN

FOR

MA

TIO

N”

(Palmgren-Miner rule)

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Gaussian random loadingsGaussian random loadings

rclccrfc hb)(1hbh

rclccrfc Hb)(1HbH

The method only works for : stationary Gaussian ((broad-band)

random loadings

Distribution of rainflow cycles :

22

21α2.11

212121app 1α

ααe)α(ααα11.112ααb2

‘rfc’ rainflow counting

‘lcc’ level-crossing counting‘rc’ range-counting

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nonnon--Gaussian random loadingsGaussian random loadingsObserved loading responses are often :• stationary (or almost-stationary)• non-Gaussian• broad-band

Gaussian : sk = ku-3 = 0

kurtosisσ

])μ(ZE[ku 4Z

4Z

skewnessσ

])μ(ZE[sk 3Z

3Z

Characterisation of non-Gaussian loading Z(t) :

OUTPUTSYSTEMSYSTEMINPUT

non-Gaussian(wave or wind loads,

road irregularity)

linearnon-Gaussian

nonlinearGaussian

EXAMPLE: data measured on a mountainEXAMPLE: data measured on a mountain--bike on offbike on off--road trackroad track

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A model for nonA model for non--Gaussian loadingsGaussian loadings

Transformed Gaussian model: Z(t) = G{ X(t) }

Inverse transformation: X(t) = g{ Z(t) }

Existing models Existing models :• Hermite (Winterstein 1988, 1994)• exponential (Ochi & Ahn, 1994)• power-law (Sarkani et al., 1994)• nonparametric (Rychlik et al., 1997)

sk=0.5 ku=5

non-Gaussian Gaussian

• time-independent (memory-less)• strictly monotonic

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G(-) is strictly monotonic :

2) xp(t1) > xp(t2) → zp(t1) > zp(t2) relative position

1) xp(ti) → zp(ti)=G{ xp(ti) } peak-peak (valley-valley) link

tit1 t2

x(t) Gaussian

z(t) non-Gaussian

rainflow count : same peak-valley couplingrainflow count : same peak-valley coupling

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Transformation of rainflow cyclesTransformation of rainflow cyclesA non-Gaussian cycle (zp , zv) will be transformed to a corresponding Gaussian cycle (xp , xv) :

zp

zvxv

xp

(xp, xv) (zp, zv) = G{ (xp,xv) } = ( G{xp}, G{xv} )

)x,(xH)g(z),g(zH)z,(zH vpG

rfc X,vpG

rfc X,vpnG

rfc Z,

g(g(--))

G(G(--))

• peaks and valleys in a random loading are random variables• transformation G(-) “shifts” probabilities

)x,(xHb)(1)x,(xHb vpG

rc X,vpG

lcc X, Gaussian case :

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Analysis schemeAnalysis scheme

NON-GAUSSIAN DATA

•Compute skew and kurt

•Estimate transformation g(-)

NONNON--GAUSSIAN DOMAINGAUSSIAN DOMAIN

g(g(--))

G(G(--))

GAUSSIAN DOMAINGAUSSIAN DOMAIN

• Estimate power spectrum

ω

G(ω)

GAUSSIAN DATA

• Estimate ‘rainflow’ distribution

rclccvpG

rfcX, Hb)(1Hb)x,(xH

• non-Gaussian ‘rainflow’ distribution

)x,(xH)z,(zH vpG

rfc X,vpnG

rfc Z,

)z,(zh vpnG

rfc Z,

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Possible analysesPossible analysesZ(t) stationary non-Gaussian loading :

neglect non-Gaussianity:

)z,(zh vpG

rfc Z,

include non-Gaussianity

)z,(zh vpnG

rfc Z,

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Case study: MountainCase study: Mountain--bike databike dataData measurements on a Mountain-bike in a Off-road use:

• various cycling conditions (uphill, downhill, level road cycling);

• different surface conditions (asphalt, cobblestone, gravel);

• both seated and standing cycling conditions.

Each measurement is clearly non-stationary.

Possible analyses:Possible analyses: -- irregularity factor, irregularity factor, IFIF-- variancevariance-- timetime--varying spectrum (STFT)varying spectrum (STFT)

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50 100 150 200 250 300 350 400 450 500 5500

0.2

0.4

0.6

0.8

1

50 100 150 200 250 300 350 400 450 500 550

-50

0

50

Time [s ]

twind = 16 secoverlap = 80 %

Irregularity factor, Irregularity factor, IFIF

50 100 150 200 250 300 350 400 450 500 5500

40

80

120

160

200

twind = 16 secoverlap = 80 %

VarianceVariance

TIME, secTIME, sec TRACKTRACK SURFACESURFACE

0 – 100 plane asphalt100 – 442 uphill gravel442 – 515 downhill cobblestn.515 – 570 plane cobblstn.+

asphalt

FORCE on the BICYCLE FORKFORCE on the FORCE on the BICYCLE FORKBICYCLE FORK

Frequency[Hz

50 100 150 200 250 300 350 400 450 500 5500

20

40

60

80

100

120

140

160

180

200

-300

-250

-200

-150

-100

-50

0

50Spectrogram (STFT)Spectrogram (STFT)

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nonnon--Gaussian dataGaussian data

Each segment is non-Gaussian

EXAMPLE EXAMPLE –– Force on bicycle forkForce on bicycle fork

50 100 150 200 250 300 350 400 450 500 550

-50

0

50

Time [s ]

Extraction of stationary segments

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Estimated fatigue cumulative spectrumEstimated fatigue cumulative spectrumComparison : experimental spectrum (from data)

non-Gaussian estimated spectrum Gaussian estimated spectrum (as if Z(t) were Gaussian).

0

10

20

30

40

50

1E-05 0.0001 0.001 0.01 0.1 1 10 100

Experimental loading spectrum

non-Gaussian estimation

Gaussian estimation

amplitude

cumulated cycles/sec

skZ = - 0.19kuZ = 4.54skX = 0.02kuX = 2.99

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0

5

10

15

20

1 10 100 1000 10000 100000

amplitude

cumulated cycles

100 blocks

Estimate fatigue life over theservice period (100 blocks )

0

5

10

15

20

1 10 100 1000

amplitude

cumulated cycles

Automotive applicationAutomotive applicationIn cooperation with

C.R.F. (Centro Ricerche FIAT)C.R.F. (Centro Ricerche FIAT)Orbassano, Italy

Stress in the critical pointfor 1 block(1 block = 60 sec)

?1 block

0

5

10

15

20

1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

observednon-GaussianGaussian

amplitude

cumulated cycles

100’000 blocks

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Analysis of nonAnalysis of non--stationarity loadingsstationarity loadings

50 100 150 200 250 300 350 400 450 500 550

-50

0

50

Time [s ]

Examples: road-induced loads in vehicles on different roads, loads in trucks switching between loaded/unloaded condition, wind/wave actions on off-shore structures under variable sea states conditions

Example of a switching loading

It is difficult to develop general models which apply to all types of load non-stationarity encountered in practical applications.

Several types of service loadings may be modelled as a sequence of adjacent stationary segments or states (“switching loadings”). Variability of switches is controlled by an underlying random process (‘regime process’).

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p

1i sii

p

1iipw dx)x(pN)s(L(s)L Ni n° rainflow cycles in i-th segment

pi(x) amplitude distribution

Loading spectrum for piece-wise variance :

Each loading spectrum Li(s) can be also estimated in the frequency-domain from PSD.

0 200 400 600 800 1000 1200 1400 1600 1800 2000-20

-10

0

10

20

time [sec]

Switching loading with constant mean valueSwitching loading with constant mean valueAdjacent load segments with:• equal mean value• constant variance• deterministic switching times

Adjacent load segments with:• equal mean value• constant variance• deterministic switching times

Benasciutti D., Tovo R.: Frequency-based fatigue analysis of non-stationary switching random loads.Fatigue Fract. Eng. Mater. Struct. 30 (2007), pp. 1-14.

s

Li(s)

segment “i”s

Lj(s)

segment “j”

...+... =loading spectrum for piece-wise variance stationary load

s

Lpw(s)

Lpw(s) = Li(s)+Lj(s)

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GOAL: Estimate the overall loading spectrum by including transition cycles.

-20

-10

0

10

20

30

40

50

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0

20

40

time [sec]

‘REGIME PROCESS’

s

L(s)

Overall loading spectrum

s

Lt(s)

PROBLEM UNDER STUDY: Switching loadings with variable mean value

GIVEN the statistical properties of:• each stationary loading segment;• the ‘regime process’.

...+... =s

Li(s)

segment “i”s

Lj(s)

segment “j”s

Lpw(s)

Lpw(s) = Li(s)+Lj(s)

Switching loading with variable mean valueSwitching loading with variable mean valueAdjacent load segments with:• different mean values• constant variance• random switching times

Adjacent load segments with:• different mean values• constant variance• random switching times

Loading spectrum for transition cycles

loading spectrum for piece-wise variance stationary load

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Numerical exampleNumerical example

100

101

102

103

104

1050

5

10

15

20

25

30

cumulated cycles

ampl

itude

from simulation Lpw(s) [transition cycles excluded]

L(s) [transition cycles included]

0

10

20

30

40

50

60

70

80

Z(t)

0 50 100 150 200 250 300 350 400 450 500

10

30

60

time [sec]

Uk

simulatedsimulated samplesample

ComparisonComparison of of loadingloading spectraspectra

5000510m3

5500010m2F =

10105000m1

m3m2m1

“From-to” matrix of ‘regime process’

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Final overview of the method Final overview of the method

multiaxial

uniaxial

Type of load

Fat Fract Eng Mat Struct (2007)"VAL 2" Conference (2009)broad-band

Gaussiannon-Gaussian

non-stationary(switching)

Int J Mat & Product Tech (2007)Fat Fract Eng Mat Struct (2009)broad-band

Gaussiannon-Gaussian

stationary

Prob Eng Mechanics (2005)Int J Fatigue (2006)broad-bandnon-Gaussian

Int J Fatigue (2002, 2005)Prob Eng Mechanics (2006)broad-bandGaussianstationary

BandwidthPDF

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ENDIFENDIFDipartimento di IngegneriaDipartimento di IngegneriaUniversità di Ferrara, ItalyUniversità di Ferrara, Italy

DIEGMDIEGMDip. Dip. IngIng. . ElettricaElettrica GestGest. . MeccanicaMeccanica

Università di Udine, ItalyUniversità di Udine, Italy

Denis Benasciutti Roberto Denis Benasciutti Roberto TovoTovo

Thanks for your attention!

[email protected]@uniud.it [email protected]@unife.it

Page 26: A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under random loading: ... rainflow count : ... Transformation of rainflow cycles A non-Gaussian

Thanks for your attention!

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Definition of the stress quantities• The Cauchy stress tensor

'xx1 2

3s 'zz

'yy2 2

1s

'xy3s '

xz4s 'yz5s

t'Itt H

3ttt2

tt

t3

ttt2t

tt3

ttt2

t'

yyxxzzzyzx

yzxxzzyy

yx

xzxyzzyyxx

ttttttttt

t

zzzyzx

yzyyyx

xzxyxx

ttr31tσH

• Deviatoric and spherical parts

• Euclidean representationof deviatoric part

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Projection by ProjectionDamage estimation

• Euclidean deviator representation

1s

3s

time

S1' S3'

time

• Projection on “principal”frame of reference

Ref erence Curve 2k

i

k2

p,ii

ρref

ρrefΓDΓD

• Partial Damage Estimation of each “projected” load history

p,iΓ

j

i,jp,ii DΓD

• Total Damage estimation by proper Partial Damage cumulating

Deperrois A. (1991)De Freitas M, Li B, Santos JLT. (2000)

Cristofori A., Susmel L., Tovo R.Int J Fatigue, Vol. 30 n. 9, pp. 1646-1658 2008