Uncertainty budget

1. Uncertainty budget • In many situations we have uncertainties come from several sources. • When the total uncertainty is too large, we look for ways of reducing it. • The tolerable total uncertainty is our uncertainty budget, and we need to achieve it by reducing individual uncertainties in the most cost effective way. • This will be illustrated by a study by Chanyoung Park on deciding between reducing uncertainties at the material level or at the structural level.

2. Modeling the Effect of Structural Tests on Uncertainty in Estimated Failure Stress (Strength) ChanyoungPark, Raphael T. Haftka, and Nam-Ho Kim

3. Multistage testing for design acceptance • Building-block process • Detect failures in early stage of design • Reduce uncertainty and estimate material properties • A large number of tests in lower pyramid (reducing uncertainty) • System-level probability of failure controlled in upper pyramid (certification) SYSTEM COMPONENTS NON-GENERIC SPECIMENS STRUCTURAL FEATURES DETAILS ELEMENTS GENERIC SPECIMENS DATA BASE COUPONS

4. Uncertainty in element strength estimates • Structural elements are under multi-axial stress and element strength has variability (aleatory uncertainty) • Element strength is estimated from material strengths in different directions using failure theory, which is not perfectly accurate (epistemic uncertainty) • Material coupon tests are done to characterize the aleatory uncertainty, but with finite number of tests we are left with errors in distributions (epistemic uncertainty) • Element tests reduce the uncertainty in failure theory. • If we can tolerate a certain total uncertainty we need to decide on number of coupon and element tests. ELEMENT COUPON

5. Estimating mean and STD of material strength • Goal: Estimate distribution of material strength from nc samples • Assumption: true material strength: tc,true~N(mc,true,sc,true) • Sample mean & STD: (mc,test, sc,test) • Predicted mean & STD tc,test tc,true tc,P tc,P ~ N(mc,P, sc,P) mc,test mc,true Distribution of distributions!!

6. Top Hat question • In tests of 50 samples, the mean strength was 100 and the standard deviation of the strength was 7. What is the typical distance between the red and purple curves in the figure • 7 • 1 • 0.7 tc,test tc,true tc,P mc,test mc,true

7. Obtaining the predictive strength distribution by sampling? • Predictive distribution of material strength tc,P ~ N(mc,P, sc,P) mc,P sc,P • Predictive mean & STD • Samples of possible material strength distributions • Predictive true material strength distribution • How do we decide how many samples? Sampling tc,P Sampling tc,P

8. Example (predictive material strength) • Predictive distribution of material strength w.r.t. # of specimens tc,P ~ N(mc,P, sc,P) tc,true tc,P nc = 30 tc,P nc = 80 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 • τc,Pis biased but it compensates by wider distribution • Are any of these results extreme compared to the expected scatter?

9. Estimating element strength • Assumption: true element strength te,true ~ N(me,true, se,true) • Error in failure theory • Element tests used to reduce errors using Bayesian updating Failure theory Coupon strength Element strength Multiaxialloading s2 te,true = ktruetc,true te,true ktrue me,true = ktruemc,true s1 tc,true Failure envelope se,true = ? me,P= (1 – ek)kcalcmc,P se,P = (1 – es)sc,P Errors

10. Prior distribution (mean element strength) me,P = (1 – ek)kcalcmc,P • Uniform distribution for error in failure theory (±10%) • Used Bayesian network to calculate PDF of me,Ptrue • Similar calculation for element std. nc mc,P mc,test sc,test me,P ek

11. Prior example • Prior distribution (Joint PDF) for the mean and STD of element strength sc,P se,P • Joint PDF me,P me,P 1.3 Uncertainty increases due to error(epistemic uncertainty) 1.2 mc,P 1.1 1.0 0.9 • 200x200 grid 0 0.05 0.1 0.15 0.2 0.25 se,P

12. Bayesian update • Element test ~ N(me,true, se,true) • Update the joint PDF (me,P, se,P) using ne element tests Reduce epistemic uncertainty in ek fM(m | test) = L(test | m)fM(m) me,P me,true

13. Illustration of convergence of coupon mc,P& sc,P • True distribution of material strength • Estimated mean of (mc,P & sc,P) (single set cumulative)

14. Illustrative example (coupon tests) • Estimated STD of (mc,P & sc,P) • Increasing nc reduces uncertainties in the estimated parameters • Effectiveness of reducing uncertainty is high at low nc

15. Top Hat question • Why does the convergence on Slide 14 (uncertainty estimates) look so much better than the convergence on Slide 13 (estimates of mean and standard deviations) • Noise to signal ratio • Difference in scales • Both

16. Illustrative element updated distribution • True distribution of element strength • Updated joint PDF of parameters • se,true • me,P • me,true • ML me,P • ML se,P • se,P

17. Element updated distributions for 10 coupon tests • RMS error (500K instances of tests) vs. uncertainty in mean and standard deviation from a single set of tests STD of mean STD of STD

18. Element updated distributions – 90 coupon tests • RMS error (500K MCS) vs. estimated uncertainty in means and standard deviation from a single set of tests. STD of mean STD of STD

19. Dependence of rms errors on number of tests Mean of element strength STD of element strength • First element test has a substantial effect to reduce uncertainty in estimated parameters of element strength