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Measuring, Using, and Reducing Experimental and Computational Uncertainty in ... · 2009-11-20 ·...
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Measuring, Using, and Reducing Experimental and Computational Uncertainty in Reliability
A l i f C i L iAnalysis of Composite Laminates
Benjamin P. SmarslokUniversity of Florida
PhD Dissertation Defense
Committee:
Dr. Raphael T. Haftka, Dr. Peter Ifju
Dr. Nam Ho Kim, Dr. Bhavani Sankar, Dr. Stanislav Uryasev
1
Motivation
• Aerospace structure’s weight and failure probability can be extremely sensitive to uncertainty
NASA X 33 R bl L h V hi l (RLV)• NASA X-33 Reusable Launch Vehicle (RLV)– Failure of composite hydrogen tanks – Residual stresses at
cryogenic temperatures caused microcracking
Hydrogen tank model
2
Outline & Objectives
• 3 observations from previous research on X 33 • 3 observations from previous research on X-33
cryogenic tanks (Qu et al. (2003) “Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments,” AIAA Journal, 41(10) 2029-2036.)
1. Failure probability is very sensitive E2 & 2 uncertainties
2. Independent material properties and observed trends
3. Significant uncertainty in estimates of small pf
• OutlineOutline1. E2 measurement uncertainty analysis and analytical 2 model for
spatial variation (not emphasized)
2 Material property correlation model V dependence2. Material property correlation model – Vf dependence
3. Separable Monte Carlo method
3
Deterministic vs. Probabilistic Design
D t i i ti th d• Deterministic method
Finite ElementFailure criteria SF
Material properties, geometry, loads, etc. Response
AnalysisFailure criteria SF
E1 , E2 , G12 , 12, w, t
1 1
2 2
12 12
,
• Probabilistic method Capacity
Finite ElementAnalysis
Limit State Function
R > Cpf
Material Properties
with uncertaintyResponse
R > C
4
Outline & Objectives – Part I
• 3 observations from previous research on X 33 • 3 observations from previous research on X-33
cryogenic tanks (Qu et al. (2003) “Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments,” AIAA Journal, 41(10) 2029-2036.)
1. Failure probability is very sensitive E2 & 2 uncertainties
2. Independent material properties and observed trends
3. Significant uncertainty in estimates of small pf
• OutlineOutline1. E2 measurement uncertainty analysis and analytical 2 model for
spatial variation (not emphasized)
2 Material property correlation model V dependence2. Material property correlation model – Vf dependence
3. Separable Monte Carlo method
5
Composite Laminates
C it l i ti• Composite lamina properties:
1
2
E1 , E2 , G12 , 12 ,
• Note: I want to introduce composite notation &
1
ote a t to t oduce co pos te otat o &background here, incomplete
Independent Properties and Trends
• In reliability-based design, the assumption of independence is often used for random variables
C id lt f Q t l (2003)• Consider results from Qu et al. (2003)
±10%±10%
• Meaning behind trends?
7
rth1
Slide 7
rth1 Needs intro that will explain that while correlation data is hard to come by, you can obtain estimates on correlation by modeling physical causes of variabiliy.You also need to discuss the two components of uncertainty, measurement errors and manufacturing variability. Raphael T. Haftka, 6/15/2009
Composite Properties vs. Vf
T d f i l hi / ( li d IM7/977 2)• Trends for a typical graphite/epoxy (applied to IM7/977-2)
• Fiber volume fraction Vf
1E2E
V V
G
fV fV
fV
(Figures from Caruso and Chamis 1986 and Rosen and Dow 1987)
12G12v
8
• Uncertainty model
Correlated Data for Material Properties
• Uncertainty model– Measurement error– Material variability
Composite propertiesX
Xe
– Composite properties:
E U t i t d l f E
T1 2 12 12, , ,E E v GX
– Ex: Uncertainty model for E2:
2 22 21 expE EE e E
true material propertyE – Total variance - covariance:
ftotal V exp
2
2 2
2 2
true material propertytrueaverageof
measured averageofexp
EE E
E E
• Measurement error is usually quantified or estimated, however variability data is often unavailable
• Use fiber volume fraction to approximate variability• Use fiber volume fraction to approximate variability
9
Composite Property Random Variables
Independent
Random Data
Correlated
Random Data
X2 X2
• Probabilistic design methods often assume independent random variables
X1 X1
• However, high correlations are expected from common physical characteristics and measurement techniques
10
q
Combining Uncertainty
• Combing material variability and measurement error:• Combing material variability and measurement error:
fV exp total where cov X X
,cov( , ) var vari j i j i jX X r X X 1 2where, cov , X X
weak correlation, exp 0.9correlated, V combined , exp .9co e ed,fV combined, total
(Correlated)(Independent)
11
Material Variability and Measurement Error in Composite Properties
• Mechanisms for composite property variability:– Fiber misalignment
Fiber packing– 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)laminateslaminates– Combine with available variance-covariance measurement error
data
• How do correlated uncertainties in composite properties propagate to strain or probability of failure?
12
rth2
Slide 12
rth2 This should appear earlier to motivate the analytical derivation of correlation Raphael T. Haftka, 6/15/2009
Glass/Epoxy: Composite Properties vs. Vf
T d f S l / f i lifi d i h i• Trends for a S-glass/epoxy from simplified micromechanics
• Fiber volume fraction range: Vf = 0.7 , = 0.025
Material Variability:1E 2E
ftotal V exp
1 12
2 12
1 1f f f m f f f m
m m
E V E V E v V v V v
E GE GE G
, , 0f fV X V Xk X fV fV
ftotal V exp1 1 1 1m m
f ff f
E GV VE G
1 0
2 0
, 1
, 2
0.033
0.07f
f
V E
V E
E
E
12G 12v
fS-glass Fiber
ComponentValue
Epoxy Matrix
Component Value
Ef (GPa) 86 Em (GPa) 4.5
12 0
12 0
, 12
, 12
0.017
0.07f
f
V v
V G
v
G
12vf 0.22 vm 0.4
Gf (GPa) 35 Gm (GPa) 1.6(from Gibson 1994)
13
fV fV
• Normally distributed fiber volume fraction
Material Variability from Fiber Volume Fraction
• Normally distributed fiber volume fraction
• Linear approximations result in properties being perfectly correlated through Vfg f rij
ftotal V exp SYM
Glass/Epoxy V = N(0 7 0 025) Graphite/Epoxy V = N(0 6 0 025)
Material Property
Mean Standard Deviation
(stdev)
Coefficient of Variation
(CV)
(G )
Glass/Epoxy Vf = N(0.7, 0.025) Graphite/Epoxy Vf = N(0.6, 0.025)
expXMaterial Property
Mean Standard Deviation
(stdev)
Coefficient of Variation
(CV)
(G )
expXE1 (GPa) 61 2.0 3.3%E2 (GPa) 21 1.5 7.0%
12 0.27 0.005 1.7%G12 (GPa) 9 9 0 69 7 0%
E1 (GPa) 150 6.4 4.25%E2 (GPa) 9.0 0.25 2.75%
12 0.34 0.005 1.5%G12 (GPa) 4 6 0 24 5 25%
14
G12 (GPa) 9.9 0.69 7.0% G12 (GPa) 4.6 0.24 5.25%
Glass/Epoxy: Measurement Uncertainty with Correlated Data
• Identify multiple properties from a single test
ftotal V exp
• Ex: Vibration testing of laminated glass/epoxy plate– Use a Bayesian statistical approach to identify joint probability
distributions of elastic constants from natural frequency q ymeasurements (Gogu et al. - SDM2009)
– Data from Pedersen and Frederiksen (1992)Data from Pedersen and Frederiksen (1992)
– 200 x 200mm plate, h = 2.5mm
– Free boundary conditions
[0 40 40 90 40 0 90 40] l– [0,-40,40,90,40,0,90,-40]s layup
15
Glass/Epoxy: Measurement Uncertainty with Correlated Data
• Vibration test identification lt
ftotal V exp
results12G
Material Property
Mean Standard Deviation
(stdev)
Coefficient of Variation
(CV)expX
1E
(stdev) (CV)
E1 (GPa) 61 1.9 3.1%E2 (GPa) 21 1.2 5.5%
12 0.27 0.03 12.2% covariance - correlation
• Use experimental data and
E1 E2 12 G12
E1 3.5e18 -0.14 -0.38 -0.633 1 17 1 4 18 0 59 0 36
12
G12 (GPa) 10 0.59 6.0%covariance - correlation
combine with correlated material variability
E2 -3.1e17 1.4e18 -0.59 -0.3612 -2.3e7 -2.3e7 1.1e-3 0.77G12 -6.9e17 -2.5e17 1.5e7 3.5e17
16
(Gogu et al. - SDM2009)
rth3
Slide 16
rth3 Again, the concept of getting correlated measurement error should appear earlier Raphael T. Haftka, 6/15/2009
Glass/Epoxy: Total Covariance & Correlation
• Total uncertainty from material variability & measurement error:
ftotal V exp
E1 E2 12 G12
E1 7.5e18 0.52 -0.35 0.29
2 7 18 3 5 18 0 46 0 46
Material Property
Coefficient of Variation
(CV)
E1 (GPa) 4.5%E2 2.7e18 3.5e18 -0.46 0.46
12 -3.2e7 -2.8e7 1.1e-3 0.39G12 7.2e17 7.8e17 1.2e6 8.3e17
1 ( )E2 (GPa) 8.9%
12 12.3%G12 (GPa) 9.2%
17
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Slide 17
rth4 It is not clear why the audience would care about these numbers and the ones on the next slide Raphael T. Haftka, 6/15/2009
Graphite/Epoxy: Material Variability
• Normal distribution for fiber volume fraction N(0.6, 0.025)
• Accuracy in prior measurements of material properties b t 1% d 2%was between 1% and 2%
– Estimate uncertainty & normal distributions
– Assume independent
Material Property
Mean
(CV) (CV) (CV)
(GP ) 150 4 25% 1% 4 4%
fV exp totalexpX
E1 (GPa) 150 4.25% 1% 4.4%E2 (GPa) 9 2.75% 3% 4.1%
12 0.34 1.5% 3% 3.4%G12 (GPa) 4 6 5 25% 3% 6 0%G12 (GPa) 4.6 5.25% 3% 6.0%
18
Summary - Comparing Total Uncertainties
Glass/Epoxy Graphite/Epoxy
Material Mean Coefficient of Variation Material Mean Coefficient
of VariationProperty of Variation(CV)
E1 (GPa) 61 4.5%E2 (GPa) 21 8.9%
Property of Variation(CV)
E1 (GPa) 150 4.4%E2 (GPa) 9 4.1%
expXexpX
12 0.27 12.3%G12 (GPa) 9.9 9.2%
12 0.34 3.4%G12 (GPa) 4.6 6.0%
E E G E E GE1 E2 12 G12
E1 7.5e18 0.52 -0.35 0.29
E2 2.7e18 3.5e18 -0.46 0.46
E1 E2 12 G12
E1 4.3e19 0.66 -0.44 0.84
E2 1.6e18 1.3e17 -0.30 0.59
12 -3.2e7 -2.8e7 1.1e-3 0.39G12 7.2e17 7.8e17 1.2e6 8.3e17
12 -3.3e7 -1.3e6 1.3e-4 -0.39G12 1.5e18 6.0e16 -1.2e6 7.7e16
19
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Slide 19
rth5 What do you want the audience to get from this avalanche of numbers? Raphael T. Haftka, 6/15/2009
Independent vs. Correlated Material Properties
Ill i l P i i• Illustrative example: Propagate uncertainty to strain
• Ex: Cylindrical pressure vessel – NASA’s X-33 RLV
4 layer (±25°) P = 100kPa (50kPa) d = 1m t = 125m– 4 layer (±25 )s P = 100kPa (50kPa) d = 1m t = 125m
0
10x HoopN
N
A
0 0y Axial
xy
N
A
2 1700 (1350 )
Max Strain Failure Criterion (deterministic)
• Compare glass/epoxy and graphite/epoxy
2max 1700 (1350 )
20
• Monte Carlo simulations (105) with correlated and
Comparison of Failure Probability
• Monte Carlo simulations (10 ) with correlated and independent properties Independent
Correlated 1
2E 0 0 0
vs.
ftotal V exp 2
12
2E
2
0 0
0
SYM
R V type
12
2G
Glass/Epoxy Graphite/EpoxyR.V. type mean(2) CV(2) pf mean(2) CV(2) pf
independent 1399 8.5% 0.010 1245 3.2% 0.007
• Not using correlations can cause an unsafe or inefficient design!
correlated 1402 7.0% 0.005 1246 4.1% 0.026
21
design!
Correlation Model Summary
U i d l i i i i• Uncertainty and correlation in composite properties:– Material variability – not always available
• Used information from fiber volume fraction
– Measurement error – usually quantified or estimated1. Correlated data
2. Experimental estimatesp
• 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!
• The effect of correlation in elastic properties on strain can The effect of correlation in elastic properties on strain can vary
22
Outline & Objectives – Part II
• 3 observations from previous research on X 33 • 3 observations from previous research on X-33
cryogenic tanks (Qu et al. (2003) “Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments,” AIAA Journal, 41(10) 2029-2036.)
1. Failure probability is very sensitive E2 & 2 uncertainties
2. Independent material properties and observed trends
3. Significant uncertainty in estimates of small pf
• OutlineOutline1. E2 measurement uncertainty analysis and analytical 2 model for
spatial variation (not emphasized)
2 Material property correlation model V dependence2. Material property correlation model – Vf dependence
3. Separable Monte Carlo method
23
• Monte Carlo simulation-based techniques can require
Probability of Failure Problems
• Monte Carlo simulation-based techniques can require expensive calculations to obtain random samples– Such as: Capacity
Limit State
Material Properties
ith t i t
Response
10
Finite ElementAnalysis
Limit State Function
R > Cpf
with uncertainty
10 NM
A BB D
• Is there a way to improve the accuracy of pf estimatewithout performing additional expensive computation?
24
Monte Carlo Simulations
M d d i d f ll b bili i• Modern structures are designed for very small probabilitiesof failure - which can have large uncertainty from simulations
• Limit state function is defined as
1 2( ) ( )R CX X capacityC , FailureS f
R CR C
where,
• Crude Monte Carlo (CMC)Most commonl sed
responseR , SafeR C
Example:– Most commonly used
1ˆN
cmc i ip I R CN
: 10, 1.25
: 13, 1.5
R N
C N
Example:
1iN
10
f
Np
0.062
25
Separable Monte Carlo Method
• If response and capacity are independent, we can look at all of the possible combinations of random samples
N M •An extension of the conditional expectation method
1 1
1ˆN M
smc i ji j
p I R CMN
N
Empirical CDF
An extension of the conditional expectation method
1
1 ˆˆ ( )N
smc C ii
p F rN
Example:
1010
f
NM
p
0 062fp 0.062
26
Monte Carlo Simulation Summary
• Crude MC traditional method for estimating pf– Looks at one-to-one evaluations of limit state
• Separable MC uses the same amount of information as CMC, but is more accurateCMC, but is more accurate– Use when limit state components are independent
– Looks at all possible combinations of limit state R.V.s
P it diff t l i f d it– Permits different sample sizes for response and capacity
27
Reliability for Bending in a Composite Plate
• Maximum deflection• Maximum deflection
• Square plate under transverse loading:allR wC w
0, sin sinx yq x y qL L
0
*q
wD
where, 4
11 12 66 224* 2 2D D D D DL
from Classical Lamination Theory (CLT)
D 4L
0
* allqD
wLimit State:
• RVs: Load, dimensions, material properties, and allowable deflection
*D
28
Using the Flexibility of Separable MC
• Plate bending random variables:[90°, 45°, -45°]s t = 125 m
Limit State:
0qR C
0
* allqD
w
• Large uncertainty in expensive response• Reformulate the problem!
29
Reformulating the Limit State
• Reduce uncertainty linked with expensive calculation
• Assume we can only afford 1,000 D* simulations
0qw
CVR CVC_____________________________
* allDw 17% 3%
7 5% 16 5%1*
allw 7.5% 16.5%
0*D q
30
Comparison of Accuracy
0 004• pf = 0.004
• Empirical variance calculated from 104 repetitions
allw w
0
* allqD
wD
1*
allwD
0*D q
31
N = 1000 (fixed) 104 reps pf = 0 004
Varying the Sample SizeN 1000 (fixed) 10 reps pf 0.004
1 allw
0*all
D q
32
Variance Estimators
• Recall crude Monte Carlo only requires an estimate of p• Recall, crude Monte Carlo only requires an estimate of pffor its variance predictor:
1ˆvar 1p p p 1 ˆ ˆ(1 )p p 1 ˆ ˆE 1 Ep p
• Separable Monte Carlo variance:
var 1cmc f fp p pN
(1 )cmc cmcp pN
E 1 Ecmc cmcp pN
2 2 21 2
11 1 1 1ˆvar E ( ) E min ,smc C f f C f
NMp F R p p F R R pN M M N M
1 2
11 ˆ ˆ ˆˆvar var cov ,smc C C C
Np F R F R F R
N N
1 1
1 N Mj i
R Ci j
I C RE F R
N M
2
2
1 1
1 N Mj i
R Ci j
I C RE F R
N M
22 1 2
1 21 1
min ,2min ,
NM
j i iR C
i j
I C R RE F R R
N M
33
Validation & Accuracy of SMC Variance EstimatesN = 1000 (fixed) 104 reps pf = 0 004N 1000 (fixed) 10 reps pf 0.004
34
Separable MC Summary
• Separable MC is a simulation-based method for pf estimates
• Inherently more accurate than crude MC
I d d t d i bl ll d t f l t th • Independent random variables allowed us to reformulate the limit state and improve accuracy of pf estimate with SMC– Desirable reduce uncertainty linked with expensive simulations in the
li it t tlimit state
– Allocate more samples to the inexpensive component
• Variance estimator was also derived:– Capable of making simulation estimates of SMC variance
2 2 21 2
11 1 1 1ˆvar E ( ) E min ,smc C f f C f
NMp F R p p F R R pN M M N M
N M M N M
35
Concluding Remarks
• Conclusions on the observations by Qu et al. (2003):– Improving techniques of thickness measurements could be a
cheap and effective way of reducing overall E uncertainty cheap and effective way of reducing overall E2 uncertainty (not covered here, in dissertation)
B i g th t i l i bilit d l b d V d – By using the material variability model based on Vf and correlated properties, then an inefficient of unsafe design could be prevented
– For statistically independent response and capacity r.v.s, separable Monte Carlo can improve accuracy of calculating p p y gpf , without much more computational cost
36
rth6
Slide 36
rth6 Second bullet should read more like: It is possible to obtain estimates of correlated variability from modeling the effects of a common cause of property variability, and it is possible to get correlated measurement errors. Raphael T. Haftka, 6/15/2009
Acknowledgements
• Financial support provided by NASA Constellation University Institute Program (CUIP)
• Dr. Laurent Carraro, Dr. David Ginsbourger, Dr. Jerome Molimard, and Dr. Rodolphe Le Riche
• Dr. Theodore F. Johnson (NASA Langley)
• Dr. Erdem Acar, Christian Gogu, Dr. Lucian Speriatu, William Schulz Bharani Ravishankar andWilliam Schulz, Bharani Ravishankar, and
Dylan Alexander (UF)
37
Separable Sampling – Tsai-Wu example
38
Variance – Covariance Review
• Independent vs correlated variance covariance matrix of • Independent vs. correlated variance - covariance matrix of lamina properties
Independent Correlated
1
2
2E
2E
0 0 0
0 0
1 1 2 1 12 1 12
2E E E E E G
2E E E2 G
p Correlated
2
12
12
E
2
2G
0
2 2 12 12
12 12 12
12
E E E2 G
2G
2G
SYMSYM
• Combine material variability and measurement error:
12G
ftotal V exp
1 2where, cov , X X
,cov( , ) var vari j i j i jX X r X X
39
Desirable Uncertainty Scenarios
• Often, the response is expensive and has large uncertainty
• Better for inexpensive capacity to have large uncertainty
• Reformulate limit state if components are independentReformulate limit state, if components are independent
• Example:– Thermal Protection System with modeling error
max1FEMBFST e T
max
1FEM
BFS eT
T
40
Simple limit state example
Assuming Response ( ) involves Expensive computation (FEA)
Limit state function
0Y 2d P
Assuming Response ( ) involves Expensive computation (FEA)• isotropic material• diameter d, thickness t • Pressure P 100 kPamax 0Y
y
zmax P
t
Failure max Y
• Pressure P= 100 kPa
x
100kPahoop Random variablesStress = f (P, d, t) ; Yield Strength, Y
axial
Stress is a linear function of load P
Regrouping the random variables
Stress is a linear function of load P
u P u – Stress per unit load
P, d, t and Y are independent random variables
4141
P Stress per unit load
Complex limit state problemDetermination of Stresses
x
y
z
Material Properties E1,E2,v12,G12
yN100kPa
Loads PLaminate Stiffness (FEA)
• Pressure vessel -1m dia
xNStiffness (FEA)
Strains • Pressure vessel -1m dia. (deterministic)
• Thickness of each lamina
Strains ,x y
Stress yxy
0.125 mm (deterministic)• Lay up- [(+25/-25)]s
Stress ,x y
Stress in each ply
x
42
s
• Internal Pressure Load,P= 100 kPa 42
Stress in each ply 1 2 12, ,
Limit State - Tsai-Wu Failure Criterion
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F
Non-separable limit state No distinct response and capacity
( , )G S
11 11 1 1
L L L L
F FS S S S
F f(Strengths S)
Random Variables
22 21 1 1
T T T T
F FS S S S
obtained from Classical Laminate Theory (CLT)
F = f(Strengths S) =f(Laminate Stiffness aij, Pressure P)
11 2266 122
.12LT
F FF F
S
F – Strength Coefficients
1 11 12
2 12 22
12 66
0 / 20 / 4
0 0 0ij
a a Pa a P a
a
PF Strength Coefficients
S – Strengths in Tension and Compression in the
Limit state G = f (F, ); G < 0 safeG ≥ 0 failed
12 660 0 0a
4343
pfiber and transverse direction
G ≥ 0 failed
Random Variables - UncertaintyCV(P ) > CV(St th ) > CV(Stiff P )
Parameters Mean CV%
CV(Pressure) > CV(Strengths) > CV(Stiffness Prop.)
E1 (GPa)
Elastic Properties (CLT)
159.1
5E2 (GPa) 8.3
G (GPa) 3 3Properties (CLT)G12 (GPa) 3.3
12 (no unit) 0.253
Pressure P Pressure P (kPa)
Load 100 15
S1T (MPa) 2312
Strengths 10
S1C (MPa) 1809
S2T (MPa) 39.2
4444
S2C (MPa) 97.2
S12 (MPa) 33.2
All the properties are assumed to have a normal distribution
( )G SOriginal limit state
Tsai – Wu Limit State Function1ˆ ( ) 0
N M
p I G S N M( , )G SOriginal limit state
Stresses Strengths SStresses per Load P
1 1
( , ) 0jsmci
ij
p I GMN
S N M
Finite Element Analysis
Expensive
From Statistical distribution
CheapFinite Element Analysis
Stresses per unit load
Load P
Cheap
From Statistical distribution
Expensive
u
Regrouping the expensive and inexpensive variables
( , )uG P,SRegrouped limit state
pp
( , )G P,SRegrouped limit state
1 1
,1ˆ ( , ) 0.
N Musm j
u
i jjicp I G
M NP
S
Expensive Cheap
Strengths SPressure Load P
Stresses per unit load u
4545
Load P
u – Material Properties, P – Pressure Loads, S – Strengths
Regrouping the random variables
St Stresses Material
PropertiesLoad
PStrengths
S
Cost Expensive Cheap Cheap
Uncertainty
~ 5% 15% 10%( )G S ( , )uG P,Sy ( , )G S ( , )G P,S
( , )G S
( , )u uG P,S
( , )u uG P,S( , )G S
4646
Comparison of the Methods 2 2 2 2 1F F F F F F ( )G S
Crude M t
Separabl M t
Separable Monte C l
11 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F
Expensive RVs- limited to N=500 (CLT)
( , )G S
SMC SMC unit load CMC
MMonte Carlo
e Monte Carlo
Carlo regrouped
RVs ˆCV cmcp ˆCV smcp ˆCV usmcp
( )Cheap RVs- varied M= 500-50000 samples
35
40
45
atio
n
SMC SMC-unit load CMC500 40.0% 20.6% 36.3%
1000 18.4% 26.0%5000 16.2% 11.7%
samples
20
25
30
ient
of V
aria 1000
016.0% 8.2%
50000
15.6% 4.0%N=500
repetitions = 10000
0
5
10
15
Coe
ffic 0 repetitions = 10000
Probability of failure,
Pf = 0.012
47
0
500 5000 50000M Samples
47
f
Summary & Conclusions
S bl M t C l t d d t bl li it t t T i Separable Monte Carlo was extended to non-separable limit state - Tsai-Wu failure criterion.
In Tsai Wu Limit State uncertainty in load affects the expensive stresses In Tsai-Wu Limit State, uncertainty in load affects the expensive stresses. By calculating response to unit loads, we can sample the effect of random loads more cheaply.
Statistical independence of the random variables enables appropriate sampling, thereby improving the accuracy of the estimate.
Shift uncertainty away from the expensive component furthers helps in accuracy gains.
Accuracy of the methods - for the same computational cost,
CMC SMC -original limit state
SMC- Regrouped limit state
4848
state state
CV% 40% 16% 4%
Relationship of pf and thickness (weight)• Qu et al (2003) “Deterministic and Reliability-Based Optimization of • Qu et al. (2003) Deterministic and Reliability-Based Optimization of
Composite Laminates for Cryogenic Environments”
• Examined the optimized X-33 tank (±25°)s for i i i i d iuncertainties sensitive to design
• Effect of improving materials on allowable strain, 2u
49
Composite Material Properties
• Temperature dependent material properties of a composite laminate:
E E G and – E1 , E2 , G12 , and 12
• Transverse modulus (E2) was observed to be sensitive to uncertainty and the focus of this work
• Hooke’s Law:• Hooke s Law:
P = LoadPE2
AA = Area
= transverse strain2
2
EA
1
A w t
50
NIST Classification of Measurement Uncertainties
• Uncertainty classification:– Random uncertainty /
variability – scatter in the variability – scatter in the measurements (v)
– Systematic uncertainty / bias –systematic departure from the systematic departure from the true value (b)
2Bb Range is at 95% (2) level
of a normal distribution
• Type of evaluation:– Type A – calculated by
statistical methods
x = experimental sample mean
vx = variability of sample
xt = true value of specimenstatistical methods
– Type B – determined by other means, such as estimate from experience
t p
x = bias error of sample
= experimental population mean
51
experience
NIST Component Measurement Uncertainty Table
P• Simplified table
– Load and transverse strain contributions were small
22
PEwt
52
Uncertainty Propagation
• Uncertainties were analyzed for all of the components of E2
• Random and systematic effects propagated separately
O l t ti t i ti h l t d ff t• Only systematic uncertainties can have correlated effects– Thickness and width are correlated
22 2 22 2 2 2 22 2 2 2( ) ( ) ( ) ( ) ( )E E E E
2 2 2 2 22 2 2 2
2 22
( ) ( ) ( ) ( ) ( )T T T TE E E EE P t wP t w
v v v v v
22 2 22 2 2 2 22 2 2 2 2 2
2 22
( ) ( ) ( ) ( ) ( ) 2 ( ) ( )T T T T T TE E E E E Eb E b P b b t b w b t b wP t w t w
• Overall uncertainty at 1 (68%) confidence for comparison
2P t w t w
to experimental results2 2
68 2 2 2( ) ( )E E EU b v
53
E2 Experimental Results
•Data is from 10 experiments at 14 temperaturesTemp
(ºC)
E2 Avg
(GPa)
CV
(%)
-168 12.5 1.05%
-145 12.24 0.87%
-126 11.96 0.84%
-78 11.46 0.81%
-53 10.76 0.88%
-28 10.22 0.98%
-3 9 82 1 07%3 9.82 1.07%
22 8.99 1.04%
48 8.63 1.16%
73 8 39 1 33%73 8.39 1.33%
99 8.18 1.48%
125 7.93 1.50%
54
151 7.51 1.73%
E2 Component Uncertainty Summary
• Thickness was the largest contributor of systematic uncertainty in E2 at 89.4% of the total bias– Surface variation contributed the most to uncertainty in thickness
• Only 30% of the observed experimental variability is from measurement uncertainty
55
Mechanisms for Composite Property Variation
• Develop a correlation model for composite material properties as a function of fiber volume fraction, Vf
U li l ti hi f i lifi d i h i • Use linear relationships of simplified micromechanics models and rule of mixtures for composite properties
• Consider carbon fiber / epoxy laminate (IM7/977-2)Consider carbon fiber / epoxy laminate (IM7/977 2)
*
N(0, 0.5)
* * * *1 120.085 0.105vf vf
E f f G f fe V V E e V V G
* 10.05 0.6f fV V
1 0 12 0
2 0 2 0
1 12
* * * *2 2
* * *12
0.055 0.6
0.03 0
E f f G f f
vf vfE f f f f
vf vfv f f f
e V V E e V V
e V V v e V
56
12 0 112v f f f
Measurement Uncertainty – Example I: Estimate
• Accuracy in prior measurements of material properties was between 1% and 2%
E ti t 2 5% & l di t ib ti• Estimate 2.5% accuracy & normal distributions– Assume independent
Correlation
SYM
57
Strains
Independent Correlated
mean()
CV()(%) pf
mean()
CV()(%) pf
Glass / Epoxy
1 1652 3.80 - 1653 4.2 -
2 1390 7.90 0.014 1392 6.8 0.009Epoxy
12 -312 35 - -311 -311 -
1 389.5 4.25 - 389.4 4.03 -
Graphite / Epoxy
2 1245 3.21 0.0073 1245 4.07 0.026
12 1020 4.82 - 1020 4.46 -
58
12 1020 4.82 1020 4.46
Combining Uncertainty in Covariance Model
• Combine material and measurement covariance
fV exp total 2
,E1 ,E1 ,E2 ,E1 , 12 ,E1 ,G12
2f f f f f f fV V V V V V V
f
,E2 ,E2 , 12 ,E2 ,G12
2, 12 , 12 ,G12
2
f f f f f
f f f
V V V V V
V V V
SYM
,G12fV 2
,E12
0 0 00 0
exp
2,E2
2, 12
2
0 00
exp
exp
SYM+ =
59
,G12exp
Vibration based identification
• Identify the four orthotropic elastic constants E1 , E2 , G12 , ν12
• Use of experimental data from Pedersen and Frederiksen (1992)• Use of experimental data from Pedersen and Frederiksen (1992)
• Measured first 10 natural frequencies of
a thin glass/epoxy composite plate
• [0,-40,40,90,40,0,90,-40]s layup
• Free boundary conditions
• Pedersen and Frederiksen used least squares to identify the elastic constants
60
Bayesian identification approach
• Bayesian approach used in current work:
– identifies a probability distribution => statistical information
– likelihood function takes uncertainty information into account– likelihood function takes uncertainty information into account
• Bayesian formulation:
posing 1 2 12 12{ , , , }E E G E
1 ( )mesure mesure mesure mesure priorf f f f f f f fE E E
Prior distribution of E
1 1 10 10 1 1 10 101... ... ( )mesure mesure mesure mesure priorf f f f f f f fK
E E E
Likelihood of the measurements given E
Posterior distribution of E given the measurements
61
Bayesian identification approach• Wide prior distribution assumed based on least squares results• Wide prior distribution assumed based on least squares results
• Likelihood function handles 2 types of uncertainty:
– Measurement error: assumed uniformly distributed with bounds at -/+1% of experimental frequencies
– Uncertainty in input parameters of the vibration model; normal uncertainties assumed:
Parameter a (mm) b (mm) h (mm) ρ (kg/m3)g
Mean 209 192 2.53 2120
St. Dev. 0.25 0.25 0.01 10.6
• Due to the propagation of uncertainties requirement, the likelihood function is very computationally expensive => use of response surface approximations (RSA)
62
approximations (RSA)
Nondimensionalization for RSA construction
• Need RSA of frequencies as a function of the relevant parameters E1 , E2 , G12 , ν12 , a, b, h and ρ
• Nondimensionalization used to construct the RSA function of smallest number of parameters, which also have physical meaning
4 4 4 4 4 2w w w w w w
• Nondimensionalizing governing eq and B C leads to:
11 16 12 66 26 224 3 2 2 3 4 24 2 2 4 0w w w w w wD D D D D D hx x y x y x y y t
• Nondimensionalizing governing eq. and B.C. leads to:
n nf 1212
11
DD
2222
11
DD
6666
11
DD
ab
4
11
haD 16
1611
DD
2626
11
DD
• Nondimensional frequency: Ψn = f(Δ12, Δ22, Δ66, Δ16, Δ26, γ)
63
MC Summary
64
MCS Variance Comparison
• Crude MC – from binomial law
1ˆvar (1 )f fp p p
• Separable MC – Condition Expectation method
var (1 )cmc f fp p pN
• Separable MC – Condition Expectation method
2 21ˆvar E ( )ce C fp F R pN
• Separable MC – Empirical CDF
N
2 2 21 2
11 1 1 1ˆvar E ( ) E min ,emp C f f C f
NMp F R p p F R R pN M M N M
65
Conditional Expectation Method
• If the one of the CDF’s are known, then we can use conditional expectation method
N
1
1ˆ ( )N
ce C ii
p F rN
1... NR r rExample:
10
f
Np
0.062
66
Separable Monte Carlo (SMC)
M h d h f h i f h d • Method gets the name from the separation of the random variables in the limit state function
( )I f d d
• Assuming that the response and capacity are independent
( , ) ( , )f C Rp I c r f c r dcdr
random variables
( ) ( )f C Rp I c r f c dc f r dr
( ) ( )f C Rp F r f r dr or 1 ( ) ( )f R Cp F c f c dc
• Separable Monte Carlo is an extension of the conditional expectation method (CE)
67
CMC Simulation Variance Estimate
• Recall, crude Monte Carlo only requires an estimate of pffor its variance predictor:
(1 )
(1 )ˆvar f f
cmc
p pp
N
• Therefore,
1
1ˆN
cmc i ii
p I c rN
ˆ ˆ(1 )ˆvar cmc cmccmc
p ppN
68
SMC Simulation Variance Estimate
• For Separable MC,
2 2 211 1 1 1ˆvar E ( ) E minNMp F R p p F R R p
3 expectations in variance equation
1 2var E ( ) E min ,emp C f f C fp F R p p F R R pN M M N M
• 3 expectations in variance equation:
1 1
1 N Mj i
R Ci j
I c rE F R
N M
1 1i j
2
2
1 1
1 N Mj i
R Ci j
I c rE F R
N M
j
22 1 2
1 21 1
min ,2min ,
NM
j i iR C
i j
I c r rE F R R
N M
69
1 1i j
Comparison of Accuracy
• pf = 3.98 x 10-3
• Empirical and estimated variance are calculated from 104
titi repetitions
1ˆ 1 ( )M
F 1ˆN M
I 1ˆ
N
p I c r 1
1 ( )ce R ii
p F cM
1 1
emp j ii j
p I c rMN
1
cmc i ii
p I c rN
70
Fc[min(R1,R2)]
71
Efficiency Comparison
• Consider an analytical example with two uniform distributions
• Use overlap ratios for simplified final expressionsp p p– Probability of r.v. being in failure region
r cR
b ap
r cC
b ap
R
12f C Rp p p
• Helpful parameter for writing a discrete expression of the variance
Rr
pR cR 2
72
variance
Analytical Example
• Crude MC
(1 )ˆvar f funi
cmc
p pp
N
ˆstdev uni
cmcp -510
• Separable MC – CE2
cmcpN
ˆd uni 75 10
• Separable MC – eCDF
2 4ˆvar 1
3funi
ceR
pp
N p
ˆstdev unicep -75×10
Separable MC eCDF
2 4 2 4ˆvar 1 1 1
3 3funi
empR C R C
pp M M N
MN p p p p
ˆstdev uniempp -77.7×10
2 24 4ˆvar 1 13 3
f f funiemp
R C
p p pp
N p M p MN
pR = 0.005 pC = 0.004 pf = 10-5 N=105 M=105
Example:
73
Simple Example - SMC Probability of Failure Grid
M = 8
: 10, 2 : 12.5, 2.5R U C U Given: pf = 0.1
N = 12
• Sort R and C and consider all possible combinations
• SMC estimate:
1 1ˆ 11 0.1158 12
N M
smc j ip I c rMN
74
1 1 8 12i jMN
Other Applications of SMC and Future Work
• Non-separable limit state (general form)
1ˆ ( ) 0N M
p I G c r 1 1
( , ) 0smc j ii j
p I G c rMN
Response Stress state Capacity Lamina strengths
11 1 66 2
1 1 1 1
L L L L LT
F F FS S S S S 11
Response – Stress state Capacity – Lamina strengths
22 21 1 1
T T T T
F FS S S S
22
12
Limit State
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F
Limit State:
75