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P. Gehl, T. Ulrich, J. Rohmer, C. Negulescu, A. Ducellier, J. Douglas

Ranking of epistemic uncertainties in scenario-based seismic risk evaluations Colloque - Modélisation des phénomènes naturels – 10/10/13. P. Gehl, T. Ulrich, J. Rohmer, C. Negulescu, A. Ducellier, J. Douglas. Projet MARS, EDF R&D. Wednesday, June 19 th , 2013. Introduction.

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P. Gehl, T. Ulrich, J. Rohmer, C. Negulescu, A. Ducellier, J. Douglas

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  1. Ranking of epistemic uncertainties in scenario-based seismic risk evaluationsColloque - Modélisation des phénomènes naturels– 10/10/13 P. Gehl, T. Ulrich, J. Rohmer, C. Negulescu, A. Ducellier, J. Douglas Projet MARS, EDF R&D Wednesday, June 19th, 2013

  2. Introduction • Large-scale risk scenarios  need for fragility curves to relate the ground motion to the level of damage • For a given building type, several sources of uncertainty: Parameter uncertainty • Strong-motion variability (seismic demand) • Structural parameters Model (structural) uncertainty • Types of analysis (e.g. capacity spectrum approach or time-history analyses) • Modelling assumptions • Objectives: • Weigh the importance of model against parameter uncertainty • Prioritize future investigations Wednesday, June 19th, 2013 >2 >2

  3. Contents • Test structure • Representation of epistemic uncertainties • Strategy for sensitivity analysis • Results and discussion Wednesday, June 19th, 2013 > 3

  4. Contents • Test structure • Representation of epistemic uncertainties • Strategy for sensitivity analysis • Results and discussion Wednesday, June 19th, 2013 > 3

  5. Test structure • Five-story RC regular building with some interior infill walls for partitioning • 3D FE model built with the OpenSees platform: • Beams and columns elements to model the frames • Truss elements for infill walls Wednesday, June 19th, 2013 > 4

  6. Fragility curve • For simplification purposes, only 2 damage states D are selected: • D1: ‘slight’ (yield) • D2: ‘near-collapse/collapse’ • Definition •  Standard Normal distribution • IM: intensity measure ~measure of the hazard (here SA(T1) , T1=0.33s) • α, β: mean and standard deviation=parameters of the fragility curve Wednesday, June 19th, 2013 > 5

  7. Contents • Test structure • Representation of epistemic uncertainties • Strategy for sensitivity analysis • Results and discussion Wednesday, June 19th, 2013 > 3

  8. Parameter uncertainty • Some mechanical parameters for steel and concrete are associated with a probabilistic distribution to account for local disparities and differences in material quality • Based on Eurocode guidelines and FEMA recommendations, some average variation ranges are proposed: • Upper/lower bounds set at +/- 2 standard deviations • This variation is only applied to frame elements (not trusses) Wednesday, June 19th, 2013 > 7

  9. Model uncertainty n°1: types of structural models • Three models are built with different sections of steel reinforcement in the first 2 stories: • Uniform steel sections  largest deformation in 1st story • Story1 x 2 and story2 x 1.5  increased participation of the 2 stories to the lateral resistance and larger deformations in 3rd story • Story1 x 2  larger deformations in 2nd story • Alteration of the deformation pattern along the height Model 1: failure of story 1 Model 2: failure of stories 1 and 3 Model 3: failure of stories 1 and 2 Wednesday, June 19th, 2013 > 8

  10. Model uncertainty n°2: Type of fragility derivation techniques • Based on the same set of ground motions: around 200 records selected from Ambraseys et al. (2004) and PEER (2001) • Capacity curve approach: • First-mode transformation of the pushover curve in the ADRS space • Ideal curve is a elastic perfectly-plastic model • Use of the unsmoothed spectra as the seismic demand • Performance point estimation is carried out with an iterative process that uses inelastic demand spectra with ductility-based reduction factors (Fajfar 1999) • Non-linear time-history analysis: • Each record is applied at the base of the model in the longitudinal direction and the maximum transient ISDR is taken Wednesday, June 19th, 2013 > 9

  11. Contents • Test structure • Representation of epistemic uncertainties • Strategy for sensitivity analysis • Results and discussion Wednesday, June 19th, 2013 > 3

  12. Variance-based global sensitivity analysis Explore the sensitivity to input parameters over their whole range of variation (i.e. in a global manner) Fully account for possible interaction between them Applicable without introducing a priori assumptions on the mathematical formulation of the landslide model

  13. Variance-based global sensitivity analysis • Basic idea: • Var(Y) = Variance of the model output Y • ~ measure of uncertainty • If I knew the true value of the input parameter X1 •  expected reduction in the variance of Var(Y) • = measure of sensitivity • Tools:Sobol’ indices (Sobol’ 1993; Saltelli et al., 2008)

  14. Sobol’ indices – in summary Variance decomposition: 2^d-1 terms ! Sensitivity index of 1st order:  Ranking in terms of importance Total sensitivity index: If 0  input parameter n°i is negligible Computation through Monte-Carlo algortihms: Number of simulations=Ns.(d+2) d # of input parameters; Ns # samples

  15. Variance-based global sensitivity analysis • Sources of uncertainty • Concrete compressive strength fc • Concrete intial Young’s modulus Ec0 • Steel yield strength fy • Steel elastic Young’s modulus Es0 • Type of structural model • Type of analysis • Use of discrete random variables (i.e. indicator Ci) to treat the case of categorical uncertainties continuous, distribution-based Categorical C1 = {1,2,3} Categorical C2 = {1,2} Wednesday, June 19th, 2013 > 12

  16. Proposed strategy • Problem definition • GSA requires a very larger number of model evaluations (~ thousands) • Duration of ONE dynamic simulation ~ 90 min Wednesday, June 19th, 2013 > 13

  17. Proposed strategy • Problem definition • GSA requires a very larger number of model evaluations (~ thousands) • Duration of ONE dynamic simulation ~ 90 min • Basic idea: Replace the computationnally intensive model G by an approximation g (= meta-model)  Costless-to-evaluateanalyticalfunction  Constructedusing a verylimitedset of input parameters’ configurations Wednesday, June 19th, 2013 > 13

  18. Meta-modelling strategy • Solution proposed : logic-tree and meta-models C2: type of analysis C1: type of structural model • Computeateachbrancha meta-model based on 40 different simulations • set of mechanicalparametersselectedthroughmaximin LHS • Select the branchthrough Monte-Carlo procedure • R²>90%; R² of cross validation ~80% Model 1 Model 2 Model 1 Model 2 Model 3 Wednesday, June 19th, 2013 > 13

  19. Smoothing Spline ANOVA Assumptions f is additive Otherorder of truncation possible! f belongs to the Second-order Sobolev set: S2={f : are absolutely continuous andL2 [0 ; 1]} x vector of d uniformly distributed variables over [0 ; 1]d n observations of the form {xi ; yi}i=1,..,n are available Wednesday, June 19th, 2013 > 13

  20. Smoothing Spline ANOVA Then the additive smoothingsplinefis the minimizer of:  controls the smoothness (chosen through CV) Fidelity to data Hastie and Tibshirani, 1990 Wednesday, June 19th, 2013 > 13

  21. Smoothing Spline ANOVA Then the additive smoothingsplinefis the minimizer of: Extreme case  controls the smoothness (chosen through CV) Fidelity to data Wednesday, June 19th, 2013 > 13

  22. Smoothing Spline ANOVA Then the additive smoothingsplinefis the minimizer of:  controls the smoothness (chosen through CV) Fidelity to data Hastie and Tibshirani, 1990 • Variable selectionthrough modification of penalty : Component Selection and ShrinkageOperator COSSO, Lin and Zhang, 2006 ; • Improvement by Storlie et al., 2010 to allowsmoothing of irrelevantcurves and more fidelity to data for important curves : Adaptive COSSO Wednesday, June 19th, 2013 > 13

  23. Contents • Test structure • Representation of epistemic uncertainties • Strategy for sensitivity analysis • Results and discussion Wednesday, June 19th, 2013 > 3

  24. Results and discussion Main effects: analysis type is the most influent for αy, whereas no source of uncertainty is dominating for αc Total effects: no source of uncertainty can be neglected for αy and αc (all > 10%) Main effects: analysis type is the most influent for βy andβc Total effects: uncertainty on mech. properties can be neglected for βy and βc (< 1 - 2%) Difference between Total and Main effects: high interactions between the input paramters for αc Number of Monte-Carlo Samples =20,000 Wednesday, June 19th, 2013 > 15

  25. Conclusion and further works • The influence of model uncertainty C2 was expected : this can serve as validation of the procedure; • Further work should focus on the integration of other model uncertainties; • The combination metamodel-ANOVA should be extented to directly integrate the discrete variables (see e.g., Storlie et al., 2013)

  26. Thank you for your attention ! Wednesday, June 19th, 2013 > 19

  27. Appendix

  28. Discussion • Fragility models are mainly influenced by the type of analysis (criterion C2) • Main effects only have been estimated for each type of analysis (light grey = static, dark grey = dynamic, I = ‘interaction terms’): • Main effects are very low for static analysis  Possible interpretation: dynamic analysis might be better able to represent each contribution via 1st-order approximations • Main effects of Ec and C1 are the highest • The effect of the type of structural model is amplified in the non-linear range (i.e. ‘collapse’ fragility parameters) Wednesday, June 19th, 2013 > 16

  29. Test structure • Mechanical properties for the constitutive models for concrete and steel: • Modal analysis  T1 = 0.33s and T2 = 0.12s (in agreement with ambient noise measurement performed on the real structure) • EMS-98 damage states are identified with ISDR, based on the yield and ultimate drift (cf. Risk-UE project) • For simplification purposes, only 2 damage states are selected: • ‘slight’: ISDR1 = 0.14% • ‘near-collapse/collapse’: ISDR2 = 0.68% Wednesday, June 19th, 2013 > 5

  30. Fragility curves • Fragility curves are estimated for each model and each analysis technique, with the standard values for mechanical parameters • SA(T1) is the selected intensity measure • Static procedure seems to over-estimate the ‘yield’ damage probability and under-estimate the ‘collapse’ one • May be explained by the bilinear model of the pushover curve: • Global strength of the building is underestimated in the elastic range • The effect of the peaks is not reproduced by the bilinear model • Standard deviation is a little greater for the static procedure Wednesday, June 19th, 2013 > 10

  31. Meta-modeling strategy • Problem: • GSA requires a very larger number of model evaluations (~ thousands) • Duration of ONE dynamic simulation ~ 90 min • Solution proposed: • Replacement of the numerical model by a meta-model approximation (Storlie et al. 2009) • Building of a meta-model as a function of the four continuous input mechanical variables, using 40 learning samples (thanks to Latin Hypercube Sampling combined to a ‘maxi-min’ space-filling design criterion) • For each of these combinations, 4 four outputs are defined: the median and standard deviation of the 2 fragility curves (‘yield’ and ‘collapse’), i.e. αy, βy, αc, βc • Use of the ACOSSO (‘Adaptive Component Selection and Shrinkage Operator’) meta-model (Storlie et al. 2010) • Procedure applied to 6 combinations of structural models and analyses (3 x 2)  6 x 4 outputs = 24 meta-models Wednesday, June 19th, 2013 > 13

  32. Computation of main and total effects • Main and total effects of each uncertainty source are estimated with the Monte-Carlo algorithm by Saltelli (2002) • 20,000 Monte-Carlo samples could be used, thanks to the various meta-models • What are main effects? • They represent the first-order contribution of each parameter (without interaction)  ranking of the influence of each parameter • What are total effects? • They include the contribution of the interaction terms  complexity of the model • A large difference between total and main effects indicates a strong non-linear influence of the parameter Wednesday, June 19th, 2013 > 14

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