Fatigue Workshop - “Broadband spectral fatigue: from Gaussian to non-Gaussian, from
research to industry”
February 24th, 2010 – Paris (F)
A comprehensive approach to fatigue
under random loading:
non-Gaussian and non-stationary loading investigations
Denis Benasciutti
DIEGM
Dip. Ing. Elettrica Gest. Meccanica
Università di Udine, Italy
Roberto Tovo
ENDIF
Dipartimento di Ingegneria
Università di Ferrara, Italy
Overview
Real service loading :
Planned research activity steps
1. Stationary, Gaussian
uniaxial loading
•
•
•
•
Random
non-Gaussian
non-stationary
multi-axial
Int J Fatigue (2002, 2005)
Prob Eng Mechanics (2006)
2. non-Gaussian loading
Prob Eng Mechanics (2005)
Int J Fatigue (2006)
3. non-stationary loading
Fat Fract Eng Mat Struct (2007)
"VAL 2" Conference (2009)
4. multi-axial loading
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)
Fatigue analysis of random loadings
TIME DOMAIN
Force \ stress \ strain
FREQUENCY DOMAIN
PSD
Time
Frequency
COUNTING METHOD
• random
• uniaxial
• stationary
(e.g. ‘rainflow’ counting)
amplitude
amplitude
CYCLE DISTRIBUTION
LOADING SPECTRUM
Force \ stress \ strain
n° cycles
Time
DAMAGE ACCUMULATION RULE
?
(e.g. Palmgren-Miner linear law)
FATIGUE LIFE
DAMAGE – FATIGUE LIFE
n° cumulated cycles (log)
Stationary random loadings
s(t)
STATIONARY LOADING
Spectral parameters :
Gaussian

λ i   ωi Gω dω
time
0
non-Gaussian
G(ω)
1 
1
0 2
; 2 
2
0 4
ω
GAUSSIAN
• narrow-band
- Rayleigh amplitude PDF
• broad-band
- 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)
Fatigue analysis of random loadings
RAINFLOW CYCLES Ci
MEASURED LOAD

+
+

For repeated measurements (in the same condition):
{C1 , C2 , ... , Cn1}
{C1 , C2 , ... , ... , Cn2}
8

{C1 , ... , Cnk}
6
4
2
0
min v

-2
-4
-6
Counted cycle: (u,v)
Max u
-8
-8
-6
-4
-2
0
2
4
6
8
Cycle distribution in random loadings
m
8
6
Isolines of h(u,v) joint PDF
h(u,v)
joint PDF
4
u v
H(u, v) 
min v
2
  h(x,y)dx dy
CDF

0
-2
u
-4
h(u,v)du dv  Prob[u, v]
-6
v
-8
-8
-6
-4
-2
0
2
Max u
u=s+m
s
s
m
v=s-m
4
6
s
8
pa,m (s,m)  2  h(s  m, s  m)
amp-mean PDF

pa (s)   pa,m (s,m) dm
-
amp. PDF
LESS “INFORMATION”
Loading spectrum and fatigue damage
PDF , CDF
h(u,v) , H(u,v)
fatigue loading spectrum
s
amp. PDF
pa(s)

F (s)   pa (x)dx
F (s)
s
fatigue damage
(Palmgren-Miner rule)
sm
D (T)  N (T) 
K
damage
D

sm 

0
sm pa (s)ds
Gaussian random loadings
Distribution of rainflow cycles :
hrfc  b hlcc  (1 b)hrc
Hrfc  b Hlcc  (1 b)Hrc
‘rfc’
rainflow counting
‘lcc’ level-crossing counting
‘rc’ range-counting

α1  α 2 1.1121  α1α 2  (α1  α 2 )  e2.11α
bapp 
α2 1 2
The method only works for :



2

 α1  α 2 
stationary
Gaussian
(broad-band)
random loadings
non-Gaussian random loadings
Observed loading responses are often :
• stationary (or almost-stationary)
• non-Gaussian
• broad-band
INPUT
Gaussian
SYSTEM
EXAMPLE: data measured on a mountainbike on off-road track
OUTPUT
nonlinear
non-Gaussian
non-Gaussian
(wave or wind loads,
road irregularity)
linear
Characterisation of non-Gaussian loading Z(t) :
E[ (Z  μZ )3 ]
sk 
σ 3Z
Gaussian :
skew ness
sk = ku-3 = 0
E[ (Z  μZ )4 ]
ku 
σ 4Z
kurtosis
A model for non-Gaussian loadings
Transformed Gaussian model:
Z(t) = G{ X(t) }
non-Gaussian
Inverse transformation:
Gaussian
(memory-less)
X(t) = g{• time-independent
Z(t) }
• strictly monotonic
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)
x(t) Gaussian
z(t) non-Gaussian
t1i
t2
G(-) is strictly monotonic :
1)
xp(ti) → zp(ti)=G{ xp(ti) }
2)
xp(t1) > xp(t2) → zp(t1) > zp(t2)
peak-peak (valley-valley) link
relative position
rainflow count : same peak-valley coupling
Transformation of rainflow cycles
zp
xp
G(-)
A non-Gaussian cycle (zp , zv) will
be transformed to a corresponding
Gaussian cycle (xp , xv) :
g(-)
xv
(xp, xv)
zv

(zp, zv) = G{ (xp,xv) } = ( G{xp}, G{xv} )
• peaks and valleys in a random loading are random variables
• transformation G(-) “shifts” probabilities
nG
Z, rfc
H
(zp, zv )  H
G
X, rfc
Gaussian case :
g(z ),g(z )  H
p
v
G
X, rfc
(xp, xv )
bHGX, lcc (xp, xv )  (1 b)HGX, rc (xp, xv )
Analysis scheme
NON-GAUSSIAN DOMAIN
GAUSSIAN DOMAIN
NON-GAUSSIAN DATA
GAUSSIAN DATA
•Compute skew and kurt
•Estimate transformation g(-)
g(-)
• Estimate power spectrum
G(ω)
ω
• non-Gaussian ‘rainflow’ distribution
G
HnG
(z
,
z
)

H
Z, rfc
p
v
X, rfc (xp , x v )
hnG
Z, rfc (zp , z v )
G(-)
• Estimate ‘rainflow’ distribution
HGX,rfc (xp, xv )  bHlcc  (1 b)Hrc
Possible analyses
Z(t)
stationary non-Gaussian loading :
 neglect non-Gaussianity:
G
Z, rfc
h
(zp, zv )
 include non-Gaussianity
nG
Z, rfc
h
(zp, zv )
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)
TIME, sec
FORCE on the
BICYCLE FORK
200
1
200
0 – 100
100 – 442
442 – 515
515 – 570
TRACK
plane
uphill
downhill
plane
Spectrogram (STFT)
Irregularity
Variance factor, IF
SURFACE
asphalt
gravel
cobblestn.
cobblstn.+
asphalt
50
180
twind = 16 sec twind = 16 sec
overlap = 80 % overlap = 80 %
Frequency [Hz]
160
0.8
160
0
140
-50
120
0.6
120
-100
100
-150
80
0.4
80
-200
60
40
0.2
40
-250
20
-300
000
50
5050
100
100
100
150
150
150
200
200
200
250
250
250
300
300
300
350
350
350
400
400
400
450
450
450
500
500500
550
550550
50
50
50
100
100
100
150
150
150
200
200
200
250
250
250
300
300
300
Time [s]
350
350
350
400
400
400
450
450
450
500
500
500
550
550
550
50
50
50
0
00
-50
-50
-50
Time [s]
0
50
100
150
200
50
100
150
200
non-Gaussian data
250
300
350
400
450
500
550
300
350
400
450
500
550
50
0
-50
250
Time [s]
Extraction of
stationary segments
EXAMPLE – Force on bicycle fork
Each segment is non-Gaussian
Estimated fatigue cumulative spectrum
Comparison :

experimental spectrum (from data)

non-Gaussian estimated spectrum
Gaussian estimated spectrum (as if Z(t) were Gaussian).

50
amplitude
Experimental loading spectrum
non-Gaussian estimation
40
Gaussian estimation
INF2-2-M-FV-SX-1
30
20
skZ = - 0.19
kuZ = 4.54
10
skX =
kuX =
0
1E-05 0.0001
0.001
0.01
0.1
1
10
100
cumulated cycles/sec
0.02
2.99
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
20
20
observed
100 blocks
1 block
amplitude
observed
amplitude
20
Estimate fatigue life over the
service period (100 blocks )
15
observed
non-Gaussian
5
10
cicli cum.
5
1 block
10
?
100
1000
cumulated cycles
100’000 blocks
0
1
10
10
5
100 blocks
0
1
ampiezza, s
ampiezza, s
10
Gaussian
15
15
100
1000
10000
100000
cumulated cycles
0
1.E+00
cicli cum.
1.E+02
1.E+04
1.E+06
1.E+08
cicli cum.
cumulated cycles
Analysis of non-stationarity loadings
1
Irregularity factor, IF
It is difficult
to develop general models which apply to all types of load non-stationarity
0.8
encountered in practical applications.
0.6
Several
types of service loadings may be modelled as a sequence of adjacent
0.4
stationary segments or states (“switching loadings”). Variability of switches is controlled
by an0.2 underlying random process (‘regime process’).
0
50
100
150
200
Example
of a switching
loading250
300
350
400
450
500
550
300
350
400
450
500
550
50
0
-50
50
100
150
200
250
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
Z(t)
Switching loading with constant mean value
20
Adjacent load segments with:
• equal mean value
10
• constant variance
0
• deterministic switching times
-10
-20
0
200
400
600
800
1000
1200
1400
1600
1800
2000
time [sec]
s
s
segment “i”
s
segment “j”
...+...
Lpw(s) = Li(s)+Lj(s)
=
Lj(s)
Li(s)
loading spectrum for
piece-wise variance
stationary load
Lpw(s)
Loading spectrum for piece-wise variance :
 

L pw (s)   Li (s)    Ni  pi ( x ) dx 
i1
i1 
s

p
p
Ni
n° rainflow cycles in i-th segment
pi(x) amplitude distribution
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. 1-14.
Switching loading with variable mean value
50
40
Loading spectrum for
transition cycles
30
s
Z(t)
20
Adjacent load segments with:
• different mean values
• constant variance
• random switching times
10
0
-10
Lt(s)
-20
Uk
40
s
Overall loading
spectrum
‘REGIME PROCESS’
20
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
loading spectrum for piecewise variance stationary load
time [sec]
s
s
segment “i”
=
...+...
Li(s)
s
segment “j”
Lj(s)
Lpw(s) = Li(s)+Lj(s)
Lpw(s)
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.
L(s)
Numerical example
80
70
“From-to” matrix of ‘regime process’
simulated sample
60
m1
m2
m3
m1
5000
10
10
m2
10
5000
5
m3
10
5
5000
Z(t)
50
40
30
20
F=
10
0
30
10
0
50
100
150
200
250
300
350
400
450
500
time [sec]
Comparison of loading spectra
from simulation
Lpw(s) [transition cycles excluded]
30
L(s) [transition cycles included]
25
amplitude
Uk
60
20
15
10
5
0 0
10
1
10
2
10
3
10
cumulated cycles
4
10
5
10
Final overview of the method
Type of load
uniaxial
stationary
non-stationary
(switching)
multiaxial
stationary
PDF
Bandwidth
Gaussian
broad-band
Int J Fatigue (2002, 2005)
Prob Eng Mechanics (2006)
non-Gaussian
broad-band
Prob Eng Mechanics (2005)
Int J Fatigue (2006)
Gaussian
non-Gaussian
broad-band
Fat Fract Eng Mat Struct (2007)
"VAL 2" Conference (2009)
Gaussian
non-Gaussian
broad-band
Int J Mat & Product Tech (2007)
Fat Fract Eng Mat Struct (2009)
Thanks for your attention!
Denis Benasciutti
denis.benasciutti@uniud.it
DIEGM
Dip. Ing. Elettrica Gest. Meccanica
Università di Udine, Italy
Roberto Tovo
roberto.tovo@unife.it
ENDIF
Dipartimento di Ingegneria
Università di Ferrara, Italy
Definition of the stress quantities
• The Cauchy stress tensor
 xx t   xy t   xz t 


 t    yx t   yy t   yz t 
 zx t   zy t   zz t 


•
Deviatoric and spherical parts
 t    H t  I   ' t 
σ H t  
 
1
tr  t 
3
 2 xx t    yy t    zz t 

 xy t 
 xz t 


3


2 yy t    zz t    xx t 

 ' t   
 yx t 
 yz t 


3

2 zz t    xx t    yy t 
 zx t 
 zy t 


3


•
s1 
Euclidean representation
of deviatoric part
3 ' s  1  '   ' 
 xx 2
yy
zz
2
2
s 3  'xy s 4   'xz
s 5  'yz
s3
•
'

Projection on “principal” 
frame of reference
S3
Euclidean deviator representation
S1
•
'
Projection by Projection
Damage estimation
s1
e
tim
•
tim
e
Cristofori A., Susmel L., Tovo R.
Int J Fatigue, Vol. 30 n. 9, pp. 1646-1658 2008
•
Total Damage estimation
by proper Partial Damage
cumulating
Γ p,i
Partial Damage Estimation of
each “projected” load history
Di Γp,i    Di,j
j
2 

k
DΓ    Di Γ p,i  ρref 
 i

k ρref
2
Re
Deperrois A. (1991)
De Freitas M, Li B,
Santos JLT. (2000)
fe
re
nc
e
Cu
rv
e