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```STATISTICAL ANALYSIS OF
FATIGUE SIMULATION DATA
J R Technical Services, LLC
140 Fairway Drive
Abingdon, Virginia
Julian Raphael
OUTLINE
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Specimen
Raw Material
Crack Growth Model
Crack Growth Parameter Variations
Simulation Results
Some Fatigue Models
Statistical Considerations
Summary
Compact Tension Specimen
Compact Specimen Dimensions
(mm)
B
W
a0
0.012 MN
0.012 MN
Validity Check
• In order that KIc is valid the following
requirement is imposed on the length of the
uncracked ligament, W-a, and the thickness,
B.
2
 K Ic 
B, W  a   2.5 
 21 mm

 
 ys 
Raw Material
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Material:
Temper:
Orientation:
Yield Strength:
Ultimate Strength:
Nominal KIc:
Cv of KIc:
Aluminum Alloy 2219
T87
L-T
393 MPa
476 MPa
36 MPa m
0.06
Crack Growth Model
 C  K 
• Crack Growth Model: Paris
• C and m Are Normally Distributed
• Nominal Paris Constants
da
dN
– C = 6.27 x 10-11
– m = 3.3
• Coefficients of Variation
– C = 0.05
– m = 0.02
m
Variations in a0 and ac
• The remaining stochastic variables are the
initial and final crack lengths, a0 and ac,
respectively.
• We assign a small coefficient of variation to
a0, because the precrack is short.
• However, ac is a function of KIc or Kc, so
the variations in KIc account for the
uncertainty in ac.
Variations in C
Normally Distributed
Variations in m
Normally Distributed
Initial Crack Length (m)
Normally Distributed
Critical Crack Length
Not Normally Distributed
Final Crack Growth Equation
• The resulting equation for Nf, the number of
fatigue cycles to fracture, with the
stochastic variables shown in red is
ac  K Ic 

Nf  
a0
da
C  K 
m
Histogram of Simulation Results
Some Statistical Concepts
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Hypothesis Test
Significance Level
Null Hypothesis
Alternate Hypothesis
P - Value
Goodness-Of-Fit
Confidence Intervals
Hypothesis Testing And The
Significance Level
• A method of making decisions using
scientific data
• A result is statistically significant if it is
unlikely to have occurred by chance alone,
according to a predetermined threshold
probability, the significance level
• Significance level is the probability of
incorrectly rejecting the null hypothesis
when it is, in fact, true, i.e. a Type I error
We Have The Failure Data, So
What Do We Do Now?
• State a null hypothesis, e.g., the data are
Normally distributed with mean m and
standard deviation . Usually denoted as H0
• State the alternate hypothesis, e.g., the data
are not Normally distributed with mean m
and standard deviation . Denoted Ha
• Pick a significance level, e.g., a  0.05
• Choose a goodness-of-fit test, e.g.,
Kolmogorov-Smirnov or Anderson-Darling
Goodness-Of-Fit Testing
• Compare your data to a standard statistical
model such as Normal. Calculate a test
statistic and compare that to a known value
that depends on the significance level and
sample size
• Goodness-of-fit tests can only tell you if
your distribution can be rejected at a
specific significance level
How Do We Choose The Best
Distribution?
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Use Goodness-Of-Fit Tests
Question - Which GOF Test is Best?
Answer - It depends on What You Want
For Example
– Kolmogorov-Smirnov gives more weight to the
center of the distribution
– Anderson-Darling gives more weight to the
tails
What Is A P-Value?
• A p-value is a probability
• Assume that your data do fit the distribution
under consideration, i.e., accept H0
temporarily
• The p-value is the probability that you
would get a goodness-of-fit statistic as
extreme or more extreme as the one you got
• P-values greater than a generally mean that
we accept the null hypothesis
PDF Comparison
CDF Comparison
Lognormal Distribution
1
f t  
e
t 2
2
ln( t ) m 


 
x
1  e
F ( x) 

2 
0
2
ln( t ) m 


2 
t
dt
m and  are the mean and standard deviation of ln(t)
Birnbaum-Saunders Distribution
 x
 
F  x    



a


x

 
1 z
 z ) 
e

2 
t2

2
dt
a is the shape factor
 is the scale factor
Confidence Intervals
• After we’ve determined what distributions cannot be
excluded it is necessary to set the ranges over which the
parameters can be expected to vary, given the sample size
• Most commonly we use 95% two-sided confidence
intervals on each parameter
• The confidence intervals depend on the mathematical
formulation of the distribution and the sample size
• The confidence interval does not mean that the parameter
will lie between the calculated values, but rather that the
true value will lie in the confidence interval 95% of the
time
Summary
• Statistical techniques are available for
distribution fitting of fatigue data
• Historically, these techniques have not been
employed frequently