G.I. SchuëllerG.I. Schuëller
Institute of Engineering MechanicsLeopold-Franzens University
Innsbruck, Austria, EU
Workshop onWorkshop onApplication of Fuzzy Sets & Application of Fuzzy Sets &
Fuzzy Logic to Engineering ProblemsFuzzy Logic to Engineering Problems
Introductory RemarksIntroductory Remarks
Pertisau, Tyrol, Austria, EUPertisau, Tyrol, Austria, EUSept. 29 – Oct. 1, 2002Sept. 29 – Oct. 1, 2002
2
Spectrum of UncertaintiesSpectrum of Uncertainties
MechanicalModel Physical
Entire Spectrum
3
Spectrum of Uncertainties - ExamplesSpectrum of Uncertainties - ExamplesWind (turbulence)
Materials (strength)
Earthquake
Buckling loads
Crack growth
4
Evolution of Structural MechanicsEvolution of Structural Mechanics
• Deterministic approachDeterministic approach
• Stochastic approachStochastic approach– Adding additional informationAdding additional information
– Replace single point by distributionReplace single point by distribution
5
Advantages of Stochastic Analysis of Advantages of Stochastic Analysis of UncertaintiesUncertainties• Quantification of Reliability possibleQuantification of Reliability possible
• More realistic response evaluationMore realistic response evaluation
• Adequate for environmental loadingAdequate for environmental loading
But:But:
• Increase in information requires increase of Increase in information requires increase of computational effortscomputational efforts
Computational Efficiency is a key Computational Efficiency is a key issueissue
( 0)fp P R S= - £
ResistanceStress
6
Quantification of RandomnessQuantification of Randomness
Random Variable Random ProcessesRandom Variable Random Processes Random Random FieldField
x1
x2
f(x1 ,x2 )
( )
1 2
1 2
1 2
1 2 1 2
( , )
,
X X
x x
F x x
f x x dx dx- ¥ - ¥
=
ò ò ( ) ( ), ,HH HHC x y C x yx x= + +
Autocorrelation Autocorrelation Function:Function:
( )1 1J PDF: ,..., ; ,...,n n nF x x t t
x
y
z
7
Random FieldsRandom Fields
• Gaussian distributedGaussian distributed
– Spectral representationSpectral representation
– Karhunen-Loéve expansionKarhunen-Loéve expansion
1 2 1 2 1 21
( , ; ) ( , ) ( ) ( , )N
jj
f x x f x x x xw z w l f=
= + ×å
1 2
1, 2 1, 1 1 2, 2 2 1, 21 2
( , ; )
2 cos( )n n n n n nn n
f x x
A x x
w
k k f¥ ¥
=- ¥ =- ¥
=
+ +å å
8
Damage Effects by Crack PropagationDamage Effects by Crack Propagation
• Structural lifeStructural life
– Crack growth is in reality a stochastic processCrack growth is in reality a stochastic process
– Deterministic modeling is just an approximationDeterministic modeling is just an approximation
• Crack growth models:Crack growth models:
– Random variablesRandom variables
– In general as SDE:In general as SDE:
( ) ( ) ,da
X g Kdt
h D= R
( )mda
C KdN
D=
R.V.R.V.
9
Bignoli et al.Bignoli et al.
Concepts to Assess UncertaintiesConcepts to Assess Uncertainties
Fuzzy Algorithm
