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)
29
Embed
A comprehensive approach to fatigue under random … · A comprehensive approach to fatigue under random loading: ... rainflow count : ... Transformation of rainflow cycles A non-Gaussian
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
- 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 :
-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
-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
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)
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
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
A model for nonA model for non--Gaussian loadingsGaussian loadings
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
Transformation of rainflow cyclesTransformation of rainflow cyclesA non-Gaussian cycle (zp , zv) will be transformed to a corresponding Gaussian cycle (xp , xv) :
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
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’).
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.
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)
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
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’
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
ENDIFENDIFDipartimento di IngegneriaDipartimento di IngegneriaUniversità di Ferrara, ItalyUniversità di Ferrara, Italy