ProcessingofFuzzySetstogetherwithRandom VariablesFUZZYSTOCHASTICANALYSISConcept{区(, t), 区(, t), x(, t), 区(, t)) → (Z(且, t), P(t))fuzzyfuzzyfuzzy randomvariablessafety levelvariablesdeterministicrandomfuzzy stochasticparametersquantitiesstructuralresponsesdepending on temporal and spatial coordinatesfuzzyanalysisα-level optimization with modified evolution strategy(in the combined space of fuzzy structural parametersand fuzzy distribution parameters)repeated stochastic analysisefficient stochasticrepresentations and Monte Carlo techniquesrepeateddeterministicstructuralanalysis,responsesurfaceapproximationMichael Beer1/7
Michael Beer 1 / 7 FUZZY STOCHASTIC ANALYSIS Concept repeated deterministic structural analysis, response surface approximation repeated stochastic analysis, efficient stochastic representations and Monte Carlo techniques fuzzy analysis, α-level optimization with modified evolution strategy (in the combined space of fuzzy structural parameters and fuzzy distribution parameters) Processing of Fuzzy Sets together with Random Variables fuzzy stochastic structural responses depending on temporal and spatial coordinates {x(θ, t), x(θ, t), X(θ, t), X(θ, t)} {Z(θ, t), Pf (t)} ~ ~ ~ → ~ • deterministic parameters fuzzy variables random quantities fuzzy random variables fuzzy safety level
Processingof FuzzySets togetherwithRandom VariablesFUZZYSTOCHASTICANALYSISExample:reliabilityanalysisPk12 =2Pu = (5/6)-Mpl1=3mMaterial:steelSt37Crosssection:rolledshapeIPE240fy: log normalP: Ex-Max-Type IFfy,min = 19.9.104 kN/m2mp=kNmf=28.8.104kN/m2Op=kNO = kN/m2MichaelBeer2/7
Michael Beer 2 / 7 FUZZY STOCHASTIC ANALYSIS Example: reliability analysis Processing of Fuzzy Sets together with Random Variables P: Ex-Max-Type I mP = kN ~ σP = kN ~ fy: log normal fy,min = 19.9·104 kN/m2 mf = 28.8·104 kN/m2 ~ σf = kN/m2 Pu = (5/6)·Mpl
Processingof FuzzySets togetherwith Random VariablesFUZZYSTOCHASTICANALYSISExample:reliabilityanalysisreliabilityindexF,(X) Aμ(B) 41.001.00.750.50?load P0.50.250.00607040505247X, [kN]0.03.1313.4133.6143.7984.113βF2(x2)1.00mp=47kN0.75mp=52kNU=1Op=4.5kNOp=6.0kN0.50-yield=Of=2.8kN/m2Of= 2.2 kN/m20.25stressf0.0022.034.0X [10°kN/m]25.826.028.830.0MichaelBeer3/7
Michael Beer 3 / 7 FUZZY STOCHASTIC ANALYSIS Example: reliability analysis Processing of Fuzzy Sets together with Random Variables load P yield stress fy reliability index mP = 52 kN σP = 6.0 kN σf = 2.8 kN/m2 mP = 47 kN σP = 4.5 kN σf = 2.2 kN/m2
ExamplesEXAMPLE1-RELIABILITYANALYSISReinforcedconcreteframev.PVOv.PVOv'Po6.00 mPH344016loadfactorfuzzy201641600'8fuzzyPH=10kNrandomPvo = 100 kNcrosssection500/35010kN/mPo=KQ18217LoadingProcesssimultaneous processing ofdead loadimprecisionanduncertainty.horizontal load PH.vertical loadsv.PvoandvPvo4/7MichaelBeer
Michael Beer 4 / 7 EXAMPLE 1 − RELIABILITY ANALYSIS 3 4 1 2 PH ν·PV 0 ν·p0 4 ø 16 4 ø 16 2 ø 16 6.00 m 8.00 Loading Process • dead load horizontal load PH • vertical loads ν·PV 0 and ν·pV 0 • ν·PV 0 PH = 10 kN PV 0 = 100 kN p0 = 10 kN/m fuzzy fuzzy random cross section 500/350 ν – load factor Reinforced concrete frame kφ kφ simultaneous processing of imprecision and uncertainty Examples
ExamplesEXAMPLE1-RELIABILITYANALYSISStructural responseSafetyLevelfuzzyfailureload(limitstate)fuzzyreliabilityindex(fuzzy-FORM)traditional,μ(β)μ(vu)req_β =3.8crisp safetylevel1.01.0:Bβ1Pr = Φ(-β)0.4β20.50.03.945.212.546.597.560.0β2 < 3.8 = req_β1≥3.8=req_βD6.437.247.63Vworstandbest case resultsforvarious intensitiesof imprecisionmaximumintensityofimprecisiontomeetreguirementsrequiredreductionofinputimprecision5/7MichaelBeer
Michael Beer 5 / 7 EXAMPLE 1 − RELIABILITY ANALYSIS 5.21 β1 ≥ 3.8 = req_β ~ β2 < 3.8 = req_β ~ β1 ~ β2 ~ 5.21 traditional, crisp safety level 6.59 7.56 β µ(β) 1.0 0.4 0.0 2.54 3.94 req_β = 3.8 β ~ µ(νu) 1.0 0.0 6.43 7.24 0.5 7.63 ν • fuzzy failure load (limit state) Safety Level • fuzzy reliability index (fuzzy-FORM) Structural response worst and best case results for various intensities of imprecision maximum intensity of imprecision to meet requirements required reduction of input imprecision Pf = Φ(−β) Examples
ExamplesEXAMPLE2-RELIABILITYANALYSISSimplesteelplatec(t,E)T=15°C,t=2.5aload,resistance,and c(t,E) randomQb(.) e [0.9,1.1]FailureprobabilityNp=45beta,CaseIbeta,Case IIbeta,Case III0.0198Nb = 2,000一interval× b() = 1.00.01990.01960.01290.01310.0128Pu0.0208E[P,]0.0126P0.0070.0110.0150.0196/7MichaelBeer
Michael Beer 6 / 7 EXAMPLE 2 − RELIABILITY ANALYSIS Simple steel plate Examples c(t,E) Q T = 15°C, t = 2.5 a load, resistance, and c(t,E) random b(.) ∈ [0.9,1.1] 0.007 0.011 0.015 0.019 Probability of Failure Beta (Case I) Beta (Case II) Beta (Case III) interval deterministic 0.0196 0.0198 0.0199 0.0208 0.0126 0.0128 0.0131 0.0129 E Pf u Pf beta, Case I beta, Case II beta, Case III interval b(.) = 1.0 0.007 0.011 0.015 0.019 Pf Failure probability Nb = 2,000 Nb = 45
ExamplesEXAMPLE2-RELIABILITYANALYSISFuzzybiasfactorFuzzyfailureprobabilityseries of. 6() = (0.8,1.0,1.2)correspondingN, = 208intervals1.01.0limit0.50.50.00.000.010.020.81.01.2 b(.)0.03Pfsensitivities of P with respect to the interval size of br(.)acceptableintervalsforparameterscanbedeterminedindications to collect additional specific informationto reducetheinputimprecisiontoanacceptablelevel7/7MichaelBeer
Michael Beer 7 / 7 EXAMPLE 2 − RELIABILITY ANALYSIS Examples Fuzzy bias factor b 081012 ( ) = . .,.,. sensitivities of PfI with respect to the interval size of bI(.) acceptable intervals for parameters can be determined • Fuzzy failure probability 0 0.5 1 0.80.8 0.9 1.01 1.1 1.21.2 b(.) µ(b) 0 0.5 1 0.000 0.010 0.020 0.030 0.0 0.5 1.0 1.0 0.5 0.0 0 0.01 0.02 0.03 Pf µ(Pf) Nb = 208 series of corresponding intervals indications to collect additional specific information to reduce the input imprecision to an acceptable level limit