measuring, using, and reducing experimental and computational uncertainty in ... · 2009-11-20 ·...

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

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

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

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

• Probabilistic method Capacity

Finite ElementAnalysis

Limit State Function

R > Cpf

Material Properties

with uncertaintyResponse

Outline & Objectives – Part I

Composite Laminates

C it l i ti• Composite lamina properties:

E1 , E2 , G12 , 12 ,

• Note: I want to introduce composite notation &

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?

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

(Figures from Caruso and Chamis 1986 and Rosen and Dow 1987)

12G12v

• Uncertainty model

Correlated Data for Material Properties

• Uncertainty model– Measurement error– Material variability

Composite propertiesX

– 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

true material propertytrueaverageof

measured averageofexp

• 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

Composite Property Random Variables

Independent

Random Data

Correlated

Random Data

• Probabilistic design methods often assume independent random variables

• However, high correlations are expected from common physical characteristics and measurement techniques

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)

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

• How do correlated uncertainties in composite properties propagate to strain or probability of failure?

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 1f f f m f f f 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

12G 12v

fS-glass Fiber

ComponentValue

Epoxy Matrix

Component Value

Ef (GPa) 86 Em (GPa) 4.5

12vf 0.22 vm 0.4

Gf (GPa) 35 Gm (GPa) 1.6(from Gibson 1994)

• 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

Glass/Epoxy Vf = N(0.7, 0.025) Graphite/Epoxy Vf = N(0.6, 0.025)

expXMaterial Property

(stdev)

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%

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

Glass/Epoxy: Measurement Uncertainty with Correlated Data

• Vibration test identification lt

ftotal V exp

results12G

Material Property

(stdev)

(CV)expX

(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

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

(Gogu et al. - SDM2009)

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

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%

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

(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%

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

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

10x HoopN

0 0y Axial

2 1700 (1350 )

Max Strain Failure Criterion (deterministic)

• Compare glass/epoxy and graphite/epoxy

2max 1700 (1350 )

• 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

ftotal V exp 2

R V type

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

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

Outline & Objectives – Part II

• 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

Finite ElementAnalysis

Limit State Function

R > Cpf

with uncertainty

A BB D

• Is there a way to improve the accuracy of pf estimatewithout performing additional expensive computation?

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

cmc i ip I R CN

: 10, 1.25

: 13, 1.5

Example:

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

1ˆN M

smc i ji j

p I R CMN

Empirical CDF

An extension of the conditional expectation method

1 ˆˆ ( )N

smc C ii

p F rN

Example:

0 062fp 0.062

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

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

where, 4

11 12 66 224* 2 2D D D D DL

from Classical Lamination Theory (CLT)

* allqD

wLimit State:

• RVs: Load, dimensions, material properties, and allowable deflection

Using the Flexibility of Separable MC

• Plate bending random variables:[90°, 45°, -45°]s t = 125 m

Limit State:

* allqD

• Large uncertainty in expensive response• Reformulate the problem!

Reformulating the Limit State

• Reduce uncertainty linked with expensive calculation

• Assume we can only afford 1,000 D* simulations

CVR CVC_____________________________

* allDw 17% 3%

7 5% 16 5%1*

allw 7.5% 16.5%

Comparison of Accuracy

0 004• pf = 0.004

• Empirical variance calculated from 104 repetitions

allw w

* allqD

N = 1000 (fixed) 104 reps pf = 0 004

Varying the Sample SizeN 1000 (fixed) 10 reps pf 0.004

1 allw

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

11 ˆ ˆ ˆˆvar var cov ,smc C C C

Np F R F R F R

1 N Mj i

R Ci j

I C RE F R

1 N Mj i

R Ci j

I C RE F R

22 1 2

1 21 1

min ,2min ,

j i iR C

I C R RE F R R

Validation & Accuracy of SMC Variance EstimatesN = 1000 (fixed) 104 reps pf = 0 004N 1000 (fixed) 10 reps pf 0.004

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

N M M N M

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

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)

Separable Sampling – Tsai-Wu example

Variance – Covariance Review

• Independent vs correlated variance covariance matrix of • Independent vs. correlated variance - covariance matrix of lamina properties

Independent Correlated

1 1 2 1 12 1 12

2E E E E E G

2E E E2 G

p Correlated

2 2 12 12

12 12 12

E E E2 G

SYMSYM

• Combine material variability and measurement error:

ftotal V exp

1 2where, cov , X X

,cov( , ) var vari j i j i jX X r X X

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

BFS eT

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

zmax P

Failure max Y

• Pressure P= 100 kPa

100kPahoop Random variablesStress = f (P, d, t) ; Yield Strength, Y

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

P Stress per unit load

Complex limit state problemDetermination of Stresses

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

• 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

F FF F

F – Strength Coefficients

1 11 12

2 12 22

0 / 20 / 4

0 0 0ij

a a Pa a P 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

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)

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

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

p I G S N M( , )G SOriginal limit state

Stresses Strengths SStresses per Load P

( , ) 0jsmci

p I GMN

Finite Element Analysis

Expensive

From Statistical distribution

CheapFinite Element Analysis

Stresses per unit load

Load P

From Statistical distribution

Expensive

Regrouping the expensive and inexpensive variables

( , )uG P,SRegrouped limit state

( , )G P,SRegrouped limit state

,1ˆ ( , ) 0.

N Musm j

i jjicp I G

Expensive Cheap

Strengths SPressure Load P

Stresses per unit load u

Load P

u – Material Properties, P – Pressure Loads, S – Strengths

Regrouping the random variables

St Stresses Material

PropertiesLoad

PStrengths

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

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

SMC SMC-unit load CMC500 40.0% 20.6% 36.3%

1000 18.4% 26.0%5000 16.2% 11.7%

samples

aria 1000

016.0% 8.2%

15.6% 4.0%N=500

repetitions = 10000

ffic 0 repetitions = 10000

Probability of failure,

Pf = 0.012

500 5000 50000M Samples

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

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

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

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

x = bias error of sample

= experimental population mean

experience

NIST Component Measurement Uncertainty Table

P• Simplified table

– Load and transverse strain contributions were small

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

( ) ( ) ( ) ( ) ( )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 ( ) ( )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

E2 Experimental Results

•Data is from 10 experiments at 14 temperaturesTemp

E2 Avg

-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%

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

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

* * * *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

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

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 -

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

,E2 ,E2 , 12 ,E2 ,G12

2, 12 , 12 ,G12

f f f f f

V V V V V

,G12fV 2

0 0 00 0

SYM+ =

,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

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

Likelihood of the measurements given E

Posterior distribution of E given the measurements

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)

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

haD 16

• Nondimensional frequency: Ψn = f(Δ12, Δ22, Δ66, Δ16, Δ26, γ)

MC Summary

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

2 2 21 2

11 1 1 1ˆvar E ( ) E min ,emp C f f C f

Conditional Expectation Method

• If the one of the CDF’s are known, then we can use conditional expectation method

1ˆ ( )N

ce C ii

p F rN

1... NR r rExample:

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)

CMC Simulation Variance Estimate

• Recall, crude Monte Carlo only requires an estimate of pffor its variance predictor:

(1 )ˆvar f f

• Therefore,

cmc i ii

p I c rN

ˆ ˆ(1 )ˆvar cmc cmccmc

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 N Mj i

R Ci j

I c rE F R

1 1i j

1 N Mj i

R Ci j

I c rE F R

22 1 2

1 21 1

min ,2min ,

j i iR C

I c r rE F R R

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

p I c r 1

1 ( )ce R ii

p F cM

emp j ii j

p I c rMN

cmc i ii

p I c rN

Fc[min(R1,R2)]

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

12f C Rp p p

• Helpful parameter for writing a discrete expression of the variance

pR cR 2

variance

Analytical Example

• Crude MC

(1 )ˆvar f funi

ˆstdev uni

cmcp -510

• Separable MC – CE2

ˆd uni 75 10

• Separable MC – eCDF

2 4ˆvar 1

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

p p pp

N p M p MN

pR = 0.005 pC = 0.004 pf = 10-5 N=105 M=105

Example:

Simple Example - SMC Probability of Failure Grid

: 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

smc j ip I c rMN

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

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:

measuring, using, and reducing experimental and computational uncertainty in ... · 2009-11-20 ·...

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