1. A comprehensive approach to fatigue under random loading:
Fatigue Workshop - “Broadband spectral fatigue: from Gaussian to non-Gaussian, from research to industry”
February 24th, – Paris (F)
A comprehensive approach to fatigue
under random loading:
non-Gaussian and non-stationary loading investigations
In this work, we present an approach to estimate the cycle distribution in a particular class of stationary random loadings.
Well, to introduce the paper topic I’ll will start with a short problem overview.
Denis Benasciutti Roberto Tovo
DIEGM
Dip. Ing. Elettrica Gest. Meccanica
Università di Udine, Italy
ENDIF
Dipartimento di Ingegneria
Università di Ferrara, Italy

2. Planned research activity steps
Overview
Real service loading :
Planned research activity steps
1. Stationary, Gaussian uniaxial loading
2. non-Gaussian loading
3. non-stationary loading
4. multi-axial loading
Int J Fatigue (2002, 2005)
Prob Eng Mechanics (2006)
Random
non-Gaussian
non-stationary
multi-axial
Prob Eng Mechanics (2005)
Int J Fatigue (2006)
Fat Fract Eng Mat Struct (2007)
"VAL 2" Conference (2009)
Int J Mat & Product Tech (2007)
Fat Fract Eng Mat Struct (2009)
This presentation:
Introduction & theoretical background
Gaussian loadings
non-Gaussian loadings. Case study: mountain-bike data, automotive application
non-stationary loadings (only a brief introduction)

3. Fatigue analysis of random loadings
TIME DOMAIN
FREQUENCY DOMAIN
Force \ stress \ strain
Time
Frequency
PSD
COUNTING METHOD
(e.g. ‘rainflow’ counting)
random
uniaxial
stationary
amplitude
amplitude
CYCLE DISTRIBUTION
LOADING SPECTRUM
Force \ stress \ strain
Time
n° cycles
n° cumulated cycles (log)
DAMAGE ACCUMULATION RULE
(e.g. Palmgren-Miner linear law) ?
FATIGUE LIFE
DAMAGE – FATIGUE LIFE

4. Stationary random loadings
STATIONARY LOADING
non-Gaussian
Spectral parameters :
Gaussian
time
G(ω) ω
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)
GAUSSIAN
narrow-band
- Rayleigh amplitude PDF
broad-band
- Wirsching & Light (1980)
- Dirlik (1985)
- Zhao & Baker (1992)
- Tovo (2002), Benasciutti & Tovo (2005)
- Markov approach (Rychlik)

5. Fatigue analysis of random loadings
MEASURED LOAD
RAINFLOW CYCLES Ci + +
  
For repeated measurements (in the same condition):
{C1 , C2 , ... , Cn1}
{C1 , C2 , ... , ... , Cn2}
Max u
min v
  
  
{C1 , ... , Cnk}
Cycle represented as dots in the (u,v) plane.
There is a “statistical law” which controls the distribution of counted cycles.
Counted cycle: (u,v)

6. Cycle distribution in random loadings
joint PDF
CDF u v
u = s + m
v = s - m m s
amp-mean PDF
amp. PDF
We describe this statistical law as a probability density function for counted cycles.
In engineering practice we refer to amplitude and mean value of a counted cycle.

7. Loading spectrum and fatigue damage
LESS “INFORMATION”
PDF , CDF
h(u,v) , H(u,v)
amp. PDF
pa(s)
damage
fatigue loading spectrum
fatigue damage
(Palmgren-Miner rule)

8. Gaussian random loadings
Distribution of rainflow cycles :
‘rfc’ rainflow counting
‘lcc’ level-crossing counting
‘rc’ range-counting
The method only works for :  stationary
 Gaussian
 (broad-band)
random loadings

9. non-Gaussian random loadings
EXAMPLE: data measured on a mountain-bike on off-road track
Observed loading responses are often :
stationary (or almost-stationary)
non-Gaussian
broad-band
OUTPUT
SYSTEM
INPUT
non-Gaussian
(wave or wind loads,
road irregularity)
linear
nonlinear
Gaussian
Gaussian : sk = ku-3 = 0
Characterisation of non-Gaussian loading Z(t) :

10. A model for non-Gaussian loadings
Transformed Gaussian model:
Z(t) = G{ X(t) }
time-independent (memory-less)
strictly monotonic
non-Gaussian
Gaussian
Inverse transformation: X(t) = g{ Z(t) }
sk=0.5 ku=5
Existing models :
Hermite (Winterstein 1988, 1994)
exponential (Ochi & Ahn, 1994)
power-law (Sarkani et al., 1994)
nonparametric (Rychlik et al., 1997)
The crucial point is to define a suitable model for the non-Gaussian loading which can be used in the fatigue analysis (that is, the estimation of rainflow cycle distribution).

11. rainflow count : same peak-valley coupling
x(t) Gaussian
z(t) non-Gaussian ti t1 t2
G(-) is strictly monotonic :
1) xp(ti) → zp(ti)=G{ xp(ti) } peak-peak (valley-valley) link
2) xp(t1) > xp(t2) → zp(t1) > zp(t2) relative position
rainflow count : same peak-valley coupling

12. Transformation of rainflow cycleszp xp
A non-Gaussian cycle (zp , zv) will be transformed to a corresponding Gaussian cycle (xp , xv) :
G(-)
g(-) xv zv
(xp, xv)  (zp, zv) = G{ (xp,xv) } = ( G{xp}, G{xv} )
peaks and valleys in a random loading are random variables
transformation G(-) “shifts” probabilities
Gaussian case :

13. Analysis scheme g(-) G(-) NON-GAUSSIAN DOMAIN GAUSSIAN DOMAIN
Estimate power spectrum ω
G(ω)
GAUSSIAN DATA
Estimate ‘rainflow’ distribution
NON-GAUSSIAN DATA
Compute skew and kurt
Estimate transformation g(-)
g(-)
non-Gaussian ‘rainflow’ distribution
G(-)

14. Possible analyses Z(t) stationary non-Gaussian loading :
 neglect non-Gaussianity:
 include non-Gaussianity

15. Case study: Mountain-bike data
Data 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: - irregularity factor, IF
- variance
- time-varying spectrum (STFT)

16. FORCE on the BICYCLE FORK
TIME, sec TRACK SURFACE
0 – plane asphalt
100 – uphill gravel
442 – downhill cobblestn.
515 – plane cobblstn.+
asphalt
FORCE on the BICYCLE FORK
Spectrogram (STFT)
twind = 16 sec
overlap = 80 %
Irregularity factor, IF
twind = 16 sec
overlap = 80 %
Variance

17. non-Gaussian data Each segment is non-Gaussian Extraction of
stationary segments
Each segment is non-Gaussian
EXAMPLE – Force on bicycle fork

18. Estimated fatigue cumulative spectrum
Comparison :  experimental spectrum (from data)
 non-Gaussian estimated spectrum
 Gaussian estimated spectrum (as if Z(t) were Gaussian).
amplitude
skZ =
kuZ = 4.54
skX = 0.02
kuX = 2.99
cumulated cycles/sec

19. Automotive application
In cooperation with
C.R.F. (Centro Ricerche FIAT)
Orbassano, Italy
Stress in the critical point
for 1 block
(1 block = 60 sec)
amplitude
amplitude
cumulated cycles
100 blocks
Estimate fatigue life over the
service period (100 blocks )
amplitude
cumulated cycles
100’000 blocks ?
1 block
cumulated cycles

20. Analysis of non-stationarity loadings
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’).
Example of a switching loading
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

21. Switching loading with constant mean value
Adjacent load segments with:
equal mean value
constant variance
deterministic switching times
loading spectrum for piece-wise variance stationary load =
Loading spectrum for piece-wise variance :
Ni n° rainflow cycles in i-th segment
pi(x) amplitude distribution
Under the simplifying assumption of an equal mean value for all load segments, the loading spectrum can be simply estimated (as a first approximation) as a linear combination of single loading spectra.
In a previous work, we showed by numerical simulations that this approximation provides quite good results.
It is worth noting that in our method we used a frequency-domain approach to estimate each loading spectrum from the PSD of each stationary load segment.
Each loading spectrum Li(s) can be also estimated in the frequency-domain from PSD.
Benasciutti D., Tovo R.: Frequency-based fatigue analysis of non-stationary switching random loads.
Fatigue Fract. Eng. Mater. Struct. 30 (2007), pp

22. Switching loading with variable mean value
Adjacent load segments with:
different mean values
constant variance
random switching times
Loading spectrum for transition cycles
Overall loading spectrum
‘REGIME PROCESS’
loading spectrum for piece-wise variance stationary load =
PROBLEM UNDER STUDY: Switching loadings with variable mean value
GIVEN the statistical properties of:
each stationary loading segment;
the ‘regime process’.
GOAL: Estimate the overall loading spectrum by including transition cycles.

23. Numerical example simulated sample Comparison of loading spectra
“From-to” matrix of ‘regime process’ m1 m2 m3
5000 10
F = 5
Comparison of loading spectra

24. Final overview of the method
Type of load
PDF
Bandwidth
uniaxial
stationary
Gaussian
broad-band
Int J Fatigue (2002, 2005)
Prob Eng Mechanics (2006)
non-Gaussian
Prob Eng Mechanics (2005)
Int J Fatigue (2006)
non-stationary
(switching)
Fat Fract Eng Mat Struct (2007)
"VAL 2" Conference (2009)
multiaxial
Int J Mat & Product Tech (2007)
Fat Fract Eng Mat Struct (2009)

25. Thanks for your attention!
Denis Benasciutti Roberto Tovo
DIEGM
Dip. Ing. Elettrica Gest. Meccanica
Università di Udine, Italy
ENDIF
Dipartimento di Ingegneria
Università di Ferrara, Italy
In this work, we present an approach to estimate the cycle distribution in a particular class of stationary random loadings.
Well, to introduce the paper topic I’ll will start with a short problem overview.

26. Thanks for your attention!

28. Definition of the stress quantities
The Cauchy stress tensor
Deviatoric and spherical parts
Euclidean representation of deviatoric part

29. Projection by Projection Damage estimation
Euclidean deviator representation
Projection on “principal” frame of reference
Cristofori A., Susmel L., Tovo R. Int J Fatigue, Vol. 30 n. 9, pp
Partial Damage Estimation of each “projected” load history
Total Damage estimation by proper Partial Damage cumulating
Deperrois A. (1991)
De Freitas M, Li B, Santos JLT. (2000)