Measuring, Using, and Reducing Experimental and Computational Uncertainty in Reliability Analysis of Composite Laminates Benjamin P. Smarslok University 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 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 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 & α2 uncertainties 2. Independent material properties and observed trends 3. Significant uncertainty in estimates of small pf (~10-8 – 10-6)

• Outline 1. E2 measurement uncertainty analysis and analytical α2 model for spatial variation (not emphasized) 2. Material property correlation model – Vf dependence 3. Separable Monte Carlo method

3

Deterministic vs. Probabilistic Design • Deterministic method Material properties, geometry, loads, etc.

E1 , E2 , G12 , ν12 α1, α2, θ, w, t

Finite Element Analysis

Response Failure criteria  ε1     ε2  , γ   12 

 σ1    σ 2  τ   12 

• Probabilistic method Material Properties with uncertainty

Finite Element Analysis

SF

Capacity

Response

Limit State Function R>C

pf

4

Outline & Objectives – Part I • 3 observations from previous research on X-33 cryogenic tanks (Qu et al. (2003) “Deterministic and ReliabilityBased Optimization of Composite Laminates for Cryogenic Environments,” AIAA Journal, 41(10) 2029-2036.) 1. Failure probability is very sensitive E2 & α2 uncertainties 2. Independent material properties and observed trends 3. Significant uncertainty in estimates of small pf

• Outline 1. E2 measurement uncertainty analysis and analytical α2 model for spatial variation (not emphasized) 2. Material property correlation model – Vf dependence 3. Separable Monte Carlo method

5

Background: Composite Laminates • Composite lamina properties: Parameter

Definition

E1 (GPa)

Longitudinal modulus

E2 (GPa)

Transverse modulus

v12

Poisson’s ratio

G12 (GPa)

Shear modulus

θ (deg)

Ply angle

h (µm)

Lamina thickness

 ε x0   Nx  −1   0   ε y  = [A]  N y  γ 0  N  xy  xy   

2

 ε1  εx       ε 2  = [T ]  ε y  γ  γ   12   xy 

1

Independent Properties and Trends • In reliability-based design, independent random variables are often used • Consider results from Qu et al. (2003)

±10% 10%

• Meaning behind trends?

7

Composite Property Random Variables

Correlated Random Data

Independent Random Data

X2

X2

X1

X1

• High correlations are expected from common physical characteristics and measurement techniques 8

Uncertainty Material Properties • Uncertainty model – Measurement error eX – Material variability υ X – Composite properties: – Ex: Uncertainty model for E2:

E2 = true material property E2 = true average of E2 E2exp = measured average of E2

• Combing correlated variability and measurement error: Συ + Σe = Σtotal cov( X , X ) = r var ( X ) var X where, Σ = cov ( X1 , X 2 )

i

j

i, j

i

( ) j

9

Correlations in Material Variability and Measurement Error • Significance of correlations has been recognized & explored – Modeling correlated input variables with copulas (Choi et al. 2009)

• However, correlations aren’t always easily available • Correlated measurement error: – Repeated identifications – Single identification with Bayesian techniques (Gogu et al. 2009) – Individual tests on same specimen

• Correlated material variability: – Unique experiments: fiber volume fraction experiments, SEM – Mechanisms: fiber misalignment, fiber packing, fiber volume fraction

Συ + Σe = Σtotal 10

Measurement Error: Graphite/Epoxy • Normal distribution for fiber volume fraction N(0.6, 0.025) • Accuracy in prior measurements of material properties was between 1% and 2% – Estimate uncertainty & normal distributions Συ + Σe = Σtotal – Assume independent Material Property

E1 (GPa) E2 (GPa)

ν12 G12 (GPa)

Mean

X

exp

150 9 0.34 4.6

Σe (CV)

1% 3% 3% 3%

11

Measurement Error with Correlated Data: Glass/Epoxy

Συ + Σe = Σtotal • Identify multiple properties from a single test • Ex: Vibration testing of laminated glass/epoxy plate – Use a Bayesian statistical approach to identify joint probability distributions of elastic constants from natural frequency measurements (Gogu et al. - SDM2009) – – – –

Data from Pedersen and Frederiksen (1992) 200 x 200mm plate, h = 2.5mm Free boundary conditions [0,-40,40,90,40,0,90,-40]s layup

12

Glass/Epoxy: Measurement Uncertainty with Correlated Data

Συ + Σe = Σtotal • Vibration test identification results Material Property

E1 (GPa) E2 (GPa)

ν12 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 correlation E1

E1

E2

ν12

G12

1

-0.14

-0.38

-0.63

1

-0.59

-0.36

1

0.77

E2

ν12

SYM

1

G12

(Gogu et al. al. - SDM2009) 13

Correlated Material Variability Outline • Develop a correlation model for composite material variability based on fiber volume fraction, Vf – Simplified micromechanics

• Consider S-glass/epoxy and carbon fiber/epoxy (IM7/977-2) laminates – Combine with available covariance measurement error data

• Compare independent and correlated material properties in an example pressure vessel problem • How do correlated uncertainties in composite properties propagate to strain or probability of failure?

Composite Properties vs. Vf • Trends for a S-glass/epoxy from simplified micromechanics • Fiber volume fraction range: Vf = 0.7 , σ = 0.025 E1 = V f E f + (1 − V f ) Em

E1

E2 =

E2

Em  E  1 − V f 1 − m  Ef  V f

v12 = V f v f + (1 − V f ) vm G12 =

Gm  G  1 − V f 1 − m G f  Vf 

G12

v12

Vf

Material Variability:

Συ + Σe = Σtotal σV

f

,X

= kV f , X X 0

σV

f

, E1

= 0.033E10

σV

f

, E2

= 0.07 E20

σV

f

,v12

= −0.017v120

σV

f

,G12

= 0.07G120

Vf 15

Material Variability from Fiber Volume Fraction • Normally distributed fiber volume fraction • Linear approximations result in properties being perfectly correlated through Vf rij

Συ + Σe = Σtotal SYM

Glass/Epoxy Vf = N(0.7, 0.025) Material Property

E1 (GPa) E2 (GPa)

ν12 G12 (GPa)

Mean

X exp

Standard Deviation (stdev)

Coefficient of Variation (CV)

61 21 0.27 9.9

2.0 1.5 0.005 0.69

3.3% 7.0% 1.7% 7.0%

Graphite/Epoxy Vf = N(0.6, 0.025) Material Property

E1 (GPa) E2 (GPa)

ν12 G12 (GPa)

Mean

X exp

Standard Deviation (stdev)

Coefficient of Variation (CV)

150 9.0 0.34 4.6

6.4 0.25 0.005 0.24

4.25% 2.75% 1.5% 5.25%

16

Summary - Comparing Total Uncertainties Glass/Epoxy Material Property

E1 (GPa) E2 (GPa)

ν12 G12 (GPa)

E1

G12

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)

ν12 G12 (GPa)

E1

E2

ν12

G12

1

0.52

-0.35

0.29

E1

1

-0.46

0.46

E2

1

0.39

ν12

1

G12

E2

ν12

Mean

Graphite/Epoxy

SYM

Mean

X exp

Coefficient of Variation (CV)

150 9 0.34 4.6

4.4% 4.1% 3.4% 6.0%

E1

E2

ν12

G12

1

0.66

-0.44

0.84

1

-0.30

0.59

1

-0.39

SYM

1

17

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 = 125µm  ε x0   N Hoop  −1   0   ε y  = [ A ]  N Axial  γ 0   0     xy  Max Strain Failure Criterion (deterministic)

ε 2max = 1700µε (1350µε ) • Compare glass/epoxy and graphite/epoxy

18

Comparison of Failure Probability • Monte Carlo simulations (105) with correlated and independent properties Independent σ E21 0  σ E22    SYM  

Correlated

Συ + Σe = Σtotal

vs.

Glass/Epoxy

0 0

σν2

12

0   0   0  σ G212 

Graphite/Epoxy

R.V. type

mean(ε2)

CV(ε2)

pf

mean(ε2)

CV(ε2)

pf

independent

1399µε

8.5%

0.010

1245µε

3.2%

0.007

correlated

1402µε

7.0%

0.005

1246µε

4.1%

0.026

• Not using correlations can cause an unsafe or inefficient design! 19

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! • The effect of correlation in elastic properties on strain can vary

20

Outline & Objectives – Part II • 3 observations from previous research on X-33 cryogenic tanks (Qu et al. (2003) “Deterministic and ReliabilityBased Optimization of Composite Laminates for Cryogenic Environments,” AIAA Journal, 41(10) 2029-2036.) 1. Failure probability is very sensitive E2 & α2 uncertainties 2. Independent material properties and observed trends 3. Significant uncertainty in estimates of small pf

• Outline 1. E2 measurement uncertainty analysis and analytical α2 model for spatial variation (not emphasized) 2. Material property correlation model – Vf dependence 3. Separable Monte Carlo method

21

Probability of Failure Problems • Monte Carlo simulation-based techniques can require expensive calculations to obtain random samples – Such as:

Capacity Response

Material Properties with uncertainty

Limit State Function R>C

Finite Element Analysis

pf

−1

ε 0   A B   N   =  M  B D κ      

Main source of computational cost

• Is there a way to improve the accuracy of pf estimate without performing additional expensive computation?

22

Monte Carlo Simulations • Limit state function is defined as C = capacity

R ( X1 ) > C ( X 2 )

R = response

where,

R > C , Failure R ≤ C , Safe

• Crude Monte Carlo (CMC) – Most commonly used

pˆ cmc

1 = N

N

∑ I [R > C ] i

i =1

i

Example: R : N (10, 1.25 ) C : N (13, 1.5 ) N = 10 p f = 0.062

23

Separable Monte Carlo Method • If response and capacity are independent, we can look at all of the possible combinations of random samples pˆ smc

1 = MN

N

M

∑∑ I  R > C  i

•An extension of the conditional expectation method

Empirical CDF

j

i =1 j =1

Example:

pˆ smc

1 = N

N

∑ Fˆ (r ) C

i

i =1

N = 10 M = 10 p f = 0.062

24

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 accurate – Use when limit state components are independent – Looks at all possible combinations of limit state R.V.s – Permits different sample sizes for response and capacity

25

Reliability for Bending in a Composite Plate • Maximum deflection

R > C → w > wall

• Square plate under transverse loading: πx  π y  q ( x, y ) = q0 sin  sin    L L     from Classical Lamination Theory (CLT)

q0 w= D* Limit State:

where,

D* =

π4

 D11 + 2 ( D12 + 2 D66 ) + D22  L 4

q0 > wall D*

• RVs: Load, dimensions, material properties, and allowable deflection 26

Using the Flexibility of Separable MC • Plate bending random variables: [90°, 45°, -45°]s t = 125 µm

Limit State:

R>C q0 > wall D*

• Large uncertainty in expensive response • Reformulate the problem!

27

Reformulating the Limit State • Reduce uncertainty linked with expensive calculation • Assume we can only afford 1,000 D* simulations CVR

CVC

_____________________________

q0 > wall D*

wall 1 > D * q0

17% 3%

7.5% 16.5%

28

Comparison of Accuracy • pf = 0.004 • Empirical variance calculated from 104 repetitions

w > wall

q0 > wall D* wall 1 > D * q0

29

Varying the Sample Size N = 1000 (fixed) 104 reps pf = 0.004

wall 1 > D * q0

30

Other Implementations of SMC • Tsai-Wu failure criterion –

Ravishankar, B., Smarslok, B.P., Haftka, R.T., and Sankar, B.V. (2009). "Separable sampling of the limit state for accurate Monte Carlo Simulation," 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Palm Springs, CA

G ( σ, F ) = F11σ 12 + F22σ 22 + F66τ 122 + F1σ 1 + F2σ 2 + 2 F12σ 1 σ 2 − 1  σ 1   A11    σ 2  =  A12 τ   0  12  

A12 A22 0

 Nx   N   y A66   0  0 0

pˆ smc

1 = MN

N

M

∑∑ I G(σ , F ) ≥ 0 i

j

i =1 j =1

• Stress failure in a ten-bar truss –

Acar et al. (2007). “Approximate probabilistic optimization using exact-capacity-approximateresponse distribution (ECARD).”

• Design optimization of an integrated thermal protection system –

Kumar et al. (2008). “Probabilistic optimization of integrated thermal protection system.”

Variance Estimators • Recall, crude Monte Carlo only requires an estimate of pf for its variance predictor: var ( pˆ cmc ) =

1 1 1 p f (1 − p f ) = E ( pˆ cmc ) (1 − E ( pˆ cmc ) ) ≈ pˆ cmc (1 − pˆ cmc ) N N N

• Separable Monte Carlo variance: var [ pˆ smc ] =

( N − 1)  1  1  M − 1  1 2 2 2    E ( ) E min , F R p p F R R p + − + − ( ) ( ) 1 2 C f  C f     M f N  M   N  M   

(

1 E R  FC ( R )  ≈ N

N

M

i =1

j =1

∑∑

I C j < Ri  M

2 E R  FC ( min ( R1 , R2 ) ) ≈ N

1 2 E R  FC ( R )  ≈   N N 2

M

i =1

j =1

∑∑

N

)

  

M

∑∑ i =1

j =1

I C j < Ri    M 

2

I C j < min ( R2 i −1 , R2 i )  M

32

Validation & Accuracy of SMC Variance Estimates N = 1000 (fixed) 104 reps pf = 0.004

33

Separable MC Summary • Separable MC is a simulation-based method for pf estimates • Inherently more accurate than crude MC • 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 limit state – Allocate more samples to the inexpensive component

• Variance estimator was also derived: – Capable of making simulation estimates of SMC variance var [ pˆ smc ] =

1 N

( N − 1)  1   M − 1  1 2 2 2    + − + − F R p p F R R p E ( ) E min , ( ) ( ) 1 2 f f C f  M   C   M   N  M    

(

)

34

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 E2 uncertainty (not covered here, in dissertation)

– It is possible to obtain estimates of correlated variability from modeling the effects of common causes of material variability (Vf) and can be combined with measurement errors – For statistically independent response and capacity RVs, separable Monte Carlo can improve accuracy of calculating pf , without much more computational cost

35

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 (EMSE) • Dr. Erdem Acar, Christian Gogu, Dr. Lucian Speriatu, William Schulz, Bharani Ravishankar, and Dylan Alexander (UF) • Dr. Theodore F. Johnson (NASA Langley)

36

Publications • Journal papers – – –

Smarslok, B.P., Haftka, R.T., Carraro, L., and Ginsbourger, D. “Improving Accuracy of Failure Probability Esitmates with Separable Monte Carlo,” Intl. J. of Reliab. and Safety (under review) Smarslok, B.P., Haftka, R.T., Ifju, P. “A Correlation Model for Composite Material Properties Combining Variability and Measurement Error” (In progress) Ravishankar, B., Smarslok, B.P., Haftka, R.T., and Sankar, B.V. "Separable Sampling of the Limit State for Accurate Monte Carlo Simulation“ (In progress)

• Conference papers –

–

–

–

–

Smarslok, B.P., Haftka, R.T., and Ifju.P. (2008). “A Correlation Model for Graphite/Epoxy Properties for Propagating Uncertainty to Strain Response,” 23rd Annual Technical Conf. of the American Society for Composites, 65, Memphis, TN Smarslok, B.P., Alexander, D., Haftka, R.T., Carraro, L., and Ginsbourger, D. (2008). “Separable Monte Carlo Applied to Laminated Composite Plates Reliability,” 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Schaumburg, IL Smarslok, B.P., Gogu, C., Haftka, R.T., and Ifju, P. (2007). “Detecting and Analyzing Nonuniformity in Mechanical and Thermal Properties of Composite Laminates,” 22nd Annual Technical Conf. of the American Society for Composites, 54, Seattle, WA Smarslok, B., Speriatu, L., Schulz, W., Haftka, R.T., Ifju, P., and Johnson, T. F. (2006). “Experimental Uncertainty in Temperature Dependent Material Properties of Composite Laminates,” Society for Experimental Mechanics Annual Conf., 241, St. Louis, MO Smarslok, B.P., Haftka, R.T., and Kim, N.H., (2006). “Taking Advantage of Separable Limit States in Sampling Procedures,” 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Newport, RI