Transcript Taking Advantage of Separable Limit States in Sampling

Correlation in elastic properties of composite
laminates due to variability in fiber volume
fraction
Based on PhD defense slides of
Benjamin P. Smarslok
Correlation model for composite elastic properties using only measurements
from a single specimen (in Press, Probabilistic Engineering Mechanics).
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Composite Laminate Elastic properties
• Composite lamina properties:
E1 , E2 , G12 , n12 , a1, a2
2
1
• Properties may have substantial correlation, but
correlation matrix may not be available
• Correlation may be able to be deduced from
dependence on fiber and matrix properties
• Here variability in fiber volume fraction is propagated to
correlation between elastic constants
Material Variability and Measurement Error in Composite
Properties
• Mechanisms for composite property variability:
– Fiber misalignment
– Fiber packing
– Fiber volume fraction
• Develop a correlation model for composite material
variability based on fiber volume fraction, Vf
• Consider S-glass/epoxy and carbon fiber/epoxy (IM7/977-2)
laminates
– Combine with available variance-covariance measurement error
data
• How do correlated uncertainties in composite properties
propagate to strain or probability of failure?
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Composite Properties vs. Vf
• Trends for a typical graphite/epoxy (applied to IM7/977-2)
• Fiber volume fraction Vf
E1
E2
Vf
Vf
Vf
G12
v12
(Figures from Caruso and
Chamis 1986 and Rosen
and Dow 1987)
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Correlated Data for Material Properties
• Uncertainty model
– Measurement error eX
– Material variability  X
– Composite properties:
X   E1 , E2 , v12 , G12 
T
– Ex: Uncertainty model for E2:


E2  1  eE2   E2 E2exp
– Total variance - covariance:
total  Vf  exp
E2  true material property
E2  true average of E2
E2exp  measured average of E2
• Measurement error is usually quantified or estimated,
however variability data is often unavailable
• Use fiber volume fraction to approximate variability
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Combining Uncertainty
• Combing material variability and measurement error:
Vf  exp  total
where,   cov  X1, X2 
weak correlation, exp
cov( X i , X j )  ri , j var  X i  var  X j 
0.9correlated, Vf


(Independent)
combined, total
(Correlated)
Is it right to add covariance matrices?
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Glass/Epoxy: Measurement Uncertainty with Correlated Data
total  Vf  exp
• Vibration test identification
results
Material
Property
E1 (GPa)
E2 (GPa)
n12
G12 (GPa)
Mean
X exp
Standard
Deviation
(stdev)
Coefficient
of Variation
(CV)
61
21
0.27
10
1.9
1.2
0.03
0.59
3.1%
5.5%
12.2%
6.0%
• Use experimental data and
combine with correlated
material variability
G12
E1
covariance - correlation
E1
E2
n12
G12
E1
3.5e18 -0.14
-0.38
-0.63
E2
-3.1e17 1.4e18 -0.59
-0.36
n12
-2.3e7 -2.3e7 1.1e-3
0.77
G12
-6.9e17 -2.5e17 1.5e7 3.5e17
(Gogu et al. - SDM2009)
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Summary - Comparing Total Uncertainties
Glass/Epoxy
Material
Property
E1 (GPa)
E2 (GPa)
n12
G12 (GPa)
Mean
Graphite/Epoxy
X exp
Coefficient
of Variation
(CV)
61
21
0.27
9.9
4.5%
8.9%
12.3%
9.2%
Material
Property
E1 (GPa)
E2 (GPa)
n12
G12 (GPa)
E1
E2
n12
G12
E1
7.5e18
0.52
-0.35
0.29
E2
2.7e18
3.5e18
-0.46
n12
-3.2e7
-2.8e7
G12
7.2e17
7.8e17
Mean
X exp
Coefficient
of Variation
(CV)
150
9
0.34
4.6
4.4%
4.1%
3.4%
6.0%
E1
E2
n12
G12
E1
4.3e19
0.66
-0.44
0.84
0.46
E2
1.6e18
1.3e17
-0.30
0.59
1.1e-3
0.39
n12
-3.3e7
-1.3e6
1.3e-4
-0.39
1.2e6
8.3e17
G12
1.5e18
6.0e16
-1.2e6
7.7e16
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Independent vs. Correlated Material Properties
• Illustrative example: Propagate uncertainty to strain
• Ex: Cylindrical pressure vessel – NASA’s X-33 RLV
– 4 layer (±25°)s P = 100kPa (50kPa) d = 1m t = 125mm
  x0 
 N Hoop 
1 
 0

  y    A   N Axial 
 0 
 0 


 xy 
Max Strain Failure Criterion (deterministic)
 2max  1700m (1350m )
• Compare glass/epoxy and graphite/epoxy
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Comparison of Failure Probability
• Monte Carlo simulations (105) with correlated and
independent properties
Independent
 E21
0

 E22


 SYM


Correlated
total  Vf  exp
vs.
Glass/Epoxy
0
0
n2
12
0 

0 

0 
 G2 12 
Graphite/Epoxy
R.V. type
mean(2)
CV(2)
pf
mean(2)
CV(2)
pf
independent
1399m
8.5%
0.010
1245m
3.2%
0.007
correlated
1402m
7.0%
0.005
1246m
4.1%
0.026
• Ignoring correlations can cause unsafe or inefficient designs!
• Is the difference in means significant?
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Correlation Model Summary
• Uncertainty and correlation in composite properties:
– Material variability – not always available
• Used information from fiber volume fraction
– Measurement error – usually quantified or estimated
1. Correlated data
2. Experimental estimates
• Combined uncertainties in a general covariance model
– Correlations don’t need to be avoided!
• Neglecting correlations by using independent RVs can
result in an inefficient or unsafe design!
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