probabilistic aspects of fatigue - illinois
TRANSCRIPT
1
Variability
Professor Darrell F. SocieDepartment of Mechanical and
Industrial Engineering
© 2003-2005 Darrell Socie, All Rights Reserved
Probabilistic Aspects of Fatigue
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Probabilistic Aspects of Fatigue
n Introductionn Basic Probability and Statisticsn Statistical Techniquesn Analysis Methods nCharacterizing VariabilitynCase Studiesn FatigueCalculator.comnGlyphWorks
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Sources of Variability
customers
materials manufacturing
usage
107
Fatigue Life, 2Nf
1
10 102 103 104 105 106
0.1
10-2
1
10-3
10-4
Str
ain
Am
plitu
de
Strength
Stress
Stress, Σ Strength, S
4
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Variability and Uncertainty
Variability: Every apple on a tree has a different mass.
Uncertainty: The variety of the apple is unknown.
Variability: Fracture toughness of a material
Uncertainty: The correct stress intensity factor solution
5
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Sources of Variability
n Stress VariablesnLoadingnCustomer UsagenEnvironment
n Strength VariablesnMaterialnProcessingnManufacturing TolerancenEnvironment
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Sources of Uncertainty
n Statistical Uncertaintyn Incomplete data (small sample sizes)
nModeling ErrornAnalysis assumptions
nHuman ErrornCalculation errorsn Judgment errors
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Modeling Variability
Products: Z = X1 • X2 • X3 • X4 • ….. Xn
Z → LogNormal as n increases
Central Limit Theorem:
COVX
0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
σ lnX σlnX ~ COVX
COVX is a good measure of variability
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Standard Deviation, lnx
COVX
1 2 3
68.3% 95.4% 99.7%
0.05
0.1
0.25
0.5
1
1.05
1.10
1.28
1.60
2.30
1.11
1.23
1.66
2.64
5.53
1.16
1.33
2.04
3.92
11.1
99.7% of the data is within a factor of ± 1.33 of the mean for a COV = 0.1
COV and LogNormal Distributions
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Variability in Service Loading
nQuantifying Loading VariabilitynMaximum LoadnLoad RangenEquivalent Stress
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Maximum Force
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
100 1000
Median 431COV 0.34
200 500
Maximum force from 42 automobile drivers99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
100 1000200 500
Cum
ulat
ive
Pro
babi
lity
Force
11
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Maximum Load Correlation
0 200 400 600 800 1000
108
107
106
105
104
103
Maximum Load
Fat
igue
Liv
es
12
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Loading Variability
1 10 102 103 104 105
Cumulative Cycles
0
16
32
48
Load
Ran
ge
54 Tractors / Drivers
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Variability in Loading
1 10
54 Tractors/Drivers
COV 0.53
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
n neq SS ∑∆=∆
Equivalent Load
Cum
ulat
ive
Pro
babi
lity
14
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Mechanisms and Slopes
100
103
104
Stre
ss A
mpl
itude
, MP
a
100
Cycles101 102 103 104 105 106 107
110
10-1210-1110-1010-910-810-710-6
Crack Growth Rate, m/cycle
1
10
100
mM
Pa
,K
∆
1
3
Crack Nucleation
Crack Growth10
100
103 104 105 106 107 108
Total Fatigue Life, Cycles
1
51
Equ
ival
ent L
oad,
kN
Structures
A combination of nucleation and growth
n = 3
n = 5
n = 10
15
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Effect of Slope on Variability
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
n = 10 5 3
Equivalent Load0.1 1 100.1 1 10
n COV3 0.535 0.4310 0.38
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Loading History Variability
n Test TracknCustomer Service
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Test Track Variability
0.1 1 10
COV 0.12
40 test track laps of a motorhome99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Equivalent Load
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Customer Usage Variability
0.1 1 10
COV 0.32
42 drivers / cars99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Equivalent Load
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Variability in Environment
n Inclusionsn Pit depth
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Inclusions That Initiated Cracks
Barter, S. A., Sharp, P. K., Holden, G. & Clark, G. “Initiation and early growth of fatigue cracks in an aerospacealuminium alloy”, Fatigue & Fracture of Engineering Materials & Structures 25 (2), 111-125.
COV = 0.27
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Pit Depth Variability
Pits12 Data PointsMedian 24.37
100
COV 0.33
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
10Pit Depth, µm
Dolly, Lee, Wei, “The Effect of Pitting Corrosion on Fatigue Life”Fatigue and Fracture of Engineering Materials and Structures, Vol. 23, 2000, 555-560
Cum
ulat
ive
Pro
babi
lity
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Variability in Materials
n Tensile Strengthn Fracture Toughnessn FatiguenFatigue StrengthnFatigue Life
n Strain-LifenCrack Growth
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Tensile Strength - 1035 Steel
25
50
75
100
500 550 600 650 700
Num
ber
of h
eats
Tensile Strength, MPa
Metals Handbook, 8th Edition, Vol. 1, p64
Mean = 602COV = 0.045
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Fracture Toughness
100
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Median 76.7
COV 0.06
60 70 9080
26 Data Points
Cum
ulat
ive
Pro
babi
lity
KIc, Ksi√in
Kies, J.A., Smith, H.L., Romine, H.E. and Bernstein, H, “Fracture Testing of Weldments”, ASTM STP 381, 1965, 328-356
Mar-M 250 Steel
25
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Fatigue Variability
102 103 104 105 106 107
Fatigue Life101
1000
100
10
1
Str
ess
Am
plitu
de Fatigue life
Fat
igue
str
engt
h
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Fatigue Life Variability
50 100 150 200 250
10
20
30
40
10
20
Num
ber
of T
ests
Life x10 350 100 150 200 250
10
20
30
40
10
20
Num
ber
of T
ests
Life x10 3
Production torsion bars5160H steelOne lot, 71 parts
25 lots, 300 parts
Metals Handbook, 8th Edition, Vol. 1, p219
µx = 123,000 cyclesCOV = 0.25
µx = 134,000 cyclesCOV = 0.27
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Statistical Variability of Fatigue Life
50
10
2
90
98
104 105 106 107 108
Cycles to Failure
Per
cent
Sur
viva
l
∆s=210∆s=245∆s=315∆s=280∆s=440
Sinclair and Dolan, “Effect of Stress Amplitude on the Variability in Fatigue Life of 7075-T6 Aluminum Alloy”Transactions ASME, 1953
7075-T6 Specimens
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COV vs Fatigue Life
440 14,000 0.12315 25,000 0.38280 220,000 0.70245 1,200,000 0.67210 12,000,000 1.39
∆S COVX
29
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Variability in Fatigue Strength
( )∏=
−+=n
1i
a2X 1C1CCOV
2i
i
085.0b)N(S2S b
f'f −≈=
∆
( ) 088.0139.11C2
'f
)085.(2S =−+=
−
30
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Strain Life Data for 950X Steel
102 103 104 105 106 107
Fatigue Life101
1
0.1
10-2
10-3
10-4
Str
ain
Am
plitu
de
378 Fatigue Tests
29 Data Sets
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29 Individual Data Sets'fσ
Median 820
300 1000
COV 0.25
2000
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
b
-0.12 -0.08 -0.04
Mean -0.09COV 0.25
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
32
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29 Individual Data Sets (continued)
0.1 1 10
Median 0.57COV 1.15
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
'fε
-0.8 -0.6 -0.4 -0.2
c
Mean -0.62COV 0.23
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
33
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Input Data Simulation
( )( ) ( ) ( )( ) ( )bb
ff
bbff ,,Ncf
'f
,,Nbf
'f N2,,LN2
E
,,L
2σµ
εεσµσσ σµε+
σµσ=
ε∆
34
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Simulation Results
0.0001
0.001
0.01
0.1
1
10
100
1 10 100 103 104 105 106 107
Stra
in A
mpl
itude
Fatigue Life
35
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Correlation
0
0.25
0.5
0.75
1.0
0.01 0.1 1 10'fε
c
0
0.05
0.1
0.15
0.2
102 103 104
'fσ
b
ρ = -0.828 ρ = -0.976
36
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Generating Correlated Data
z1 = ( rand() )Φz2 = ( rand() )Φ
2213 1zzz ρ−+ρ=
)zexp( 1lnln'f '
f'f σσ σ+µ=σ
3bb zb σ+µ=
z1= N(0,1)
37
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Correlated Properties
0.0001
0.001
0.01
0.1
1
10
100
1 10 100 103 104 105 106 107
Fatigue Life
Stra
in A
mpl
itude
38
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Curve Fitting
102 103 104 105 106 107
Fatigue Life101
1
0.1
10-2
10-3
10-4
Str
ain
Am
plitu
de
cf
'f
bf
'f )N2()N2(
E2ε+
σ=
ε∆
Assume a constant slope to get a distribution of properties
'fσ
39
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Property Distribution
0.1 1
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
365 Data Points
Median 0.26
COV 0.42
'fε
378 Data Points
Median 802 MPa
COV 0.12
'fσ
100 1000
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
b = -0.086 c = -0.51
40
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Correlation
0
500
1000
1500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7'fε
'fσ
41
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Simulation
0.001
0.01
0.1
1
10
0.00011 10 100 103 104 105 106 107
Stra
in A
mpl
itude
Fatigue Life
42
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Strength Coefficient
5000
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
500
0
500
1000
1500
0 500 1000 1500 2000 2500
365 Data Points
Median 1002 MPa
COV 0.14
'fσ
'K
'K
ρ = 0.863
43
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Crack Growth Data
0
10
20
30
40
50
Cra
ck L
engt
h, m
m
0 50 100 150 200 250 300 350
Cycles x103
Virkler, Hillberry and Goel, “The Statistical Nature of Fatigue Crack Propagation”, Journal of Engineering Materials
and Technology, Vol. 101, 1979, 148-153
44
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Crack Growth Rate Data
300,000
68 Data Points
Median 257,000COV 0.07
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Fatigue Lives Crack Growth Rate
5 10 5010-5
10-4
10-3
10-2C
rack
gro
wth
rate
, mm
/cyc
le
Stress intensity, MPa√m
45
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Crack Growth Properties
Median 5 x 10-8
COV 0.44
10-7
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
10-6
Median 3.13COV 0.06
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
2 54
mKCdNda
∆=
C m
46
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Beware of Correlated Variables
( )2/m1SC
aaN
2m
m
2/m1i
2/m1f
f
−π∆
−=
−−
Nf and C are linearly related and should have the same variability, but
07.0COVfN =
44.0COVC =
because C and m are correlated.
47
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Correlation
2.0
2.5
3.0
3.5
4.0
10-8 10-610-7
C
m
48
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Correlated Variables
2YYX
2XZ 2 σ+σρσ+σ=σ
-1 -0.5 0 0.5 1
0.5
1
1.5
2
ρ
σZ
σX = σY = 1
49
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Calculated Lives
105 106
Computed fromC and m pairs experimental
105 106
0.1 %0.1 %
99.9 %
99 %
90 %
50 %
10 %
1 %
Cum
ulat
ive
Pro
babi
lity
99.9 %
99 %
90 %
50 %
10 %
1 %
Cum
ulat
ive
Pro
babi
lity
experimental
( )∫ ∆=
f
o
a
amf
KCda
N
50
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Manufacturing/Processing Variability
n Bolt Forcesn Surface FinishnDrilled Holes
51
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Variability in Bolt Force
100 1000
Force200 Data PointsMedian 130COV 0.14
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Bolt Force, kN
Preload force in bolts tightened to 350 Nm
Cum
ulat
ive
Pro
babi
lity
52
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Surface Roughness Variability
100
125 Data PointsMedian 43COV 0.10
Machined aluminum casting
10Surface Finish, µin
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Surface Finish
53
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Drilled HolesFighter Spectrum
154 Data Points
Median 126,750
COV 0.22 in life
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
105 Cycles
From: J.P. Butler and D.A. Rees, "Development of Statistical Fatigue Failure Characteristics of 0.125-inch 2024-T3 Aluminum Under Simulated Flight-by-Flight Loading," ADA-002310 (NTIS no.), July 1974.
180 drilled holes in a single plate
COV 0.07 in strength
54
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Analysis Uncertainty
nMiners Linear Damage rulen Strain Life Analysis
55
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50
90
99.9
99
10
1
0.01
0.01 0.1 1 10 100
Computed Life / Experimental
Cum
ulat
ive
Pro
babi
lity
of F
ailu
re 964 Tests
COV = 1.02
Miners Rule
From Erwin Haibach “ Betriebsfestigkeit”, Springer -Verlag, 2002
A safety factor of 10 in life would result in a 10% chance of failure
56
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SAE Specimen
Suspension
Transmission
Bracket
Fatigue Under Complex Loading: Analysis and Experiments, SAE AE6, 1977
57
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Analysis Results
1 10 100
48 Data Points
COV 1.27
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Analytical Life / Experimental Life
Strain-Life analysis of all test data
58
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Material Variability
1 10 100
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
Analytical Life / Experimental Life
Material Analysis
Strain-Life back calculation of specimen lives
59
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Modeling Uncertainty
( )∏=
−+=n
1i
a2X 1C1CCOV
2i
i
CU = 1.09
Analysis Uncertainty CU = ?
The variability in reproducing the original strain life data from the material constants is CM ~ 0.44
2M
2N2
U C1
C1C1 f
+
+=+
90% of the time the analysis is within a factor of 3 !99% of the time the analysis is within a factor of 10 !
60
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Variability from Multiple Sources
( )∏=
−+=n
1i
a2X 1C1CCOV
2i
i
Suppose we have 4 variables each with a COV = 0.1
The combined variability is COV = 0.29
Suppose we reduce the variability of one of the variables to 0.05
The combined variability is now COV = 0.27
If all of the COV’s are the same, it doesn’t do any good to reduce only one of them, you must reduce all of them !
61
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Variability from Multiple Sources
( )∏=
−+=n
1i
a2X 1C1CCOV
2i
i
Suppose we have 3 variables each with a COV = 0.1and one with COV = 0.4
The combined variability is COV = 0.65
Suppose we reduce the variability of these variables to 0.05
The combined variability is now COV = 0.60
If one of the COV’s is large, it doesn’t do any good to reduce the others, you must reduce the largest one !
62
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Variability Summary
Source COVService Loading 0.5
Materials 0.1Manufacturing 0.1Surface Finish 0.1
Environment 0.3
Strength
Stress
Fatigue LivesAnalysis Uncertainty
1.01.0
5
StressStrength
lifeFatigue
∝
63
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Variability and Uncertainty
Variability: Every apple on a tree has a different mass.
Uncertainty: The variety of the apple is unknown.
Variability: Multiple samples of the same material
Uncertainty: What is the material
64
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Strain Life Data for 93 Steels
1 10 102 103 106 10710510410-4
10-2
10-3
0.1
10
1
Str
ain
Am
plitu
de
1 10 102 103 106 10710510410-4
10-2
10-3
0.1
10
1
Str
ain
Am
plitu
de
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Uncertainty for all Steels
100 1000 10000
55 Data Points
COV 0.48
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
'fσ
66
5 Variability © 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 68
Uncertainty for Structural Steels
100 1000 10000
COV 0.12
99.9 %
99 %
90 %
50 %
10 %
1 %
0.1 %
Cum
ulat
ive
Pro
babi
lity
'fσ Su 600 – 900 MPa
19 Data Points
67
5 Variability © 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 68
Variability and Uncertainty
Fatigue Strength Coefficient
Variability Uncertainty CombinedAll Steels 0.12 0.48 0.75
Structural Steel 0.12 0.12 0.24
68
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Quiz
What is the expected variability ?
At my last seminar everyone hit a golf ball and we recorded the maximum acceleration.
69
5 Variability © 2003-2005 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 68
Results
12.97.7
5.8811.115.510.318.111.644.260.37
µσ
COV
70
Probabilistic Aspects of Fatigue