UncertaintyQuantificationLeibniztilMichaelBeer102UniversitatInstituteforRiskand Reliability1o0:4Hannover
Engineering & Uncertainty Analysis Part I: Engineering Mathematics − Numerical Solution of Engineering Problems Uncertainty Quantification Michael Beer Institute for Risk and Reliability
ReliabilityandRiskAnalysisLeibniz102UniversitatGENERALSITUATIONto04HannoverEndeavornumericalmodeling-physicalphenomena,structure,andenvironmentprognosis-systembehavior,hazards,safety,risk,robustness,economic and social impact,...closeto realitynumericallyefficientDeterministicmethodsdeterministicdeterministicRealitycomputationalmodelsstructural parametersUncertaintyImprecisionVagueness?Fluctuations?Ambiguity ?ConsequencesVariability?Indeterminacy?2/29
2 / 29 Reliability and Risk Analysis GENERAL SITUATION » close to reality » numerically efficient Endeavor • numerical modeling − physical phenomena, structure, and environment prognosis − system behavior, hazards, safety, risk, robustness, economic and social impact, . Deterministic methods deterministic structural parameters Reality deterministic computational models • • Uncertainty ? Variability ? Consequences ? Vagueness ? Indeterminacy ? Fluctuations ? Imprecision ? Ambiguity ?
ReliabilityandRiskAnalysisLeibniz10:2UniversitatUncertaintiesto04HannoverGeneralstatement,There is nothing so wrong with the analysisas believing the answer!"RichardP.FeynmanClassification of error-related problemsblunders/grosserrors:stronglymisleadingresultsexample:linear analysiscomputationaltargetof a stronglyresultsnonlinearprobleminaccuracy/bias:systematicdeviationfromtheactualvaluecomputationalexample: negligence of a certain minoreffect8target-resultsinanapproximationsolutionimprecision:spread around theactual valueuncertaintyscattercomputationalexample:randomeffectsinthespecificationtargetofinputquantitiessuchasamaterialstrengthresultsStatistics and Probability Theory3/29
3 / 29 Reliability and Risk Analysis Uncertainties General statement „There is nothing so wrong with the analysis as believing the answer!“ Richard P. Feynman Classification of error-related problems target computational results example: linear analysis of a strongly nonlinear problem target computational results example: negligence of a certain minor effect in an approximation solution ● inaccuracy / bias: systematic deviation from the actual value ● blunders / gross errors: strongly misleading results target computational results example: random effects in the specification of input quantities such as a material strength ● uncertainty / imprecision: scatter / spread around the actual value Statistics and Probability Theory
ReliabilityandRiskAnalysisLeibniz10:2UniversitatUncertaintiesto04HannoverExample:compressivestrengthofconcreteetakeasamplefromafreshconcretemixture.form 20specimensof egualgeometrystorethespecimensforhardening over28 days under equal conditions.testthe cylinder compressive strengthf.samplestrengthsamplestrengthX=f,[N/mm2]X=f[N/mm2]elementelement11122.3426.20Which value to use21226.6126.40for calculations?31326.6820.6141424.5925.79Howriskyis itto51523.9625.30selecta certainvalue?61630.3726.0871724.1028.2781822.6823.20Can wefindacertain91920.3924.94valuecorrespondingto201025.7425.60an acceptablerisk?4/29
4 / 29 Reliability and Risk Analysis Can we find a certain value corresponding to an acceptable risk ? Example: compressive strength of concrete ● take a sample from a fresh concrete mixture ● form 20 specimens of equal geometry ● store the specimens for hardening over 28 days under equal conditions ● test the cylinder compressive strength fc sample strength element x=fc [N/mm2] 1 22.34 2 26.40 3 20.61 4 24.59 5 25.30 6 26.08 7 24.10 8 22.68 9 24.94 10 25.60 sample strength element x=fc [N/mm2] 11 26.20 12 26.61 13 26.68 14 25.79 15 23.96 16 30.37 17 28.27 18 23.20 19 20.39 20 25.74 Which value to use for calculations ? How risky is it to select a certain value ? Uncertainties
ReliabilityandRiskAnalysisiLeibniz102UniversitatUncertaintiesto04HannoverMission of statistics and probability theory.real datasummaryofthedescriptivetypical properties of the datastatistics(e.g.,magnitudeand scatter)drawconclusionsregardingtheinferentialunderlyinggeneration scheme of thedata,modeling and guantification of uncertaintystatistics(e.g.,certain"form"of the scatter)processing oftheuncertaintycalculation of uncertain quantitiesprobabilityforthe results as a basis to derive decisionstheory(e.g.,scatter of a deflection,failureprobabilities)Stochastics.decisions(e.g.,design,strengthening)fromGreekandLatin:NOTE:Statisticscannotmakedecisions!"theart of conjecturingIthelpsustomakedecisions.5/29
5 / 29 Reliability and Risk Analysis Mission of statistics and probability theory ● real data descriptive statistics inferential statistics probability theory ● summary of the typical properties of the data (e.g., magnitude and scatter) ● draw conclusions regarding the underlying generation scheme of the data, modeling and quantification of uncertainty (e.g., certain "form" of the scatter) ● processing of the uncertainty, calculation of uncertain quantities for the results as a basis to derive decisions (e.g., scatter of a deflection, failure probabilities) ● decisions (e.g., design, strengthening) NOTE: Statistics cannot make decisions ! It helps US to make decisions. Stochasticsmathematical statistics from Greek and Latin: "the art of conjecturing" Uncertainties