1. G.I. Schuëller Institute of Engineering Mechanics Leopold-Franzens University Innsbruck, Austria, EU Workshop on Application of Fuzzy Sets & Fuzzy Logic to Engineering Problems Introductory Remarks Pertisau, Tyrol, Austria, EU Sept. 29 – Oct. 1, 2002

2. 2 Spectrum of Uncertainties Mechanical Model Physical Entire Spectrum

3. 3 Spectrum of Uncertainties - Examples Wind (turbulence) Materials (strength) Earthquake Buckling loads Crack growth

4. 4 Evolution of Structural Mechanics Deterministic approachDeterministic approach Stochastic approachStochastic approach –Adding additional information –Replace single point by distribution

5. 5 Advantages of Stochastic Analysis of Uncertainties Quantification of Reliability possibleQuantification of Reliability possible More realistic response evaluationMore realistic response evaluation Adequate for environmental loadingAdequate for environmental loadingBut: Increase in information requires increase of computational effortsIncrease in information requires increase of computational efforts Computational Efficiency is a key issue Resistance Stress

6. 6 Quantification of Randomness Random Variable Random Processes Random Field Random Variable Random Processes Random Field x1x1 x2x2 f(x 1,x 2 ) Autocorrelation Function: x y z

7. 7 Random Fields Gaussian distributedGaussian distributed –Spectral representation –Karhunen-Loéve expansion

8. 8 Damage Effects by Crack Propagation Structural lifeStructural life –Crack growth is in reality a stochastic process –Deterministic modeling is just an approximation Crack growth models:Crack growth models: –Random variables –In general as SDE: R.V.

9. 9 Bignoli et al. Concepts to Assess Uncertainties Fuzzy Algorithm