Institute for Statics und Dynamics of Structures Fuzzy probabilistic safety assessment

Transcription

1 Institute for Statics und Dynamics of Structures Fuzzy probabilistic safety assessment Bernd Möller

2 Fuzzy probabilistic safety assessment randomness fuzzy randomness fuzzyness failure Fuzzy- Probabilistik failure robability P probability P f f P f = x... f (x) X ; x f X s dx Slide 2 of 45

3 Fuzzy probabilistic safety assessment Methods for computing the fuzzy failure probability P f Computing of P f original by original. If one original from each fuzzy probability basic variable is known, the assigned failure probability may be calculated, e.g., by approximated integration. Fuzzy-Monte-Carlo Simulation Improving the numerical efficiency by: Importance Sampling Subset Sampling Line Sampling Fuzzy first order reliability - FORM Slide 3 of 45

4 Original space of the basic variables data uncertainty Y f(x) fuzzy design point x B model uncertainty Y g(x) = 0 A x 2 µ = 1 fuzzy joint probability density function f(x) µ = 0 1 µ(x B ) fuzzy limit state surface g(x) = 0 s fuzzy survival region g(x) > 0 µ = 0 µ = 1 x B t fuzzy failure region g(x) < 0 Slide 4 of 45

5 Fuzzy first order reliability method g(x) α=1 > 0 x 2 g(x) α=1 = 0 g(x) α=1 = 0 g(x) α=1 # 0 x 2 µ(x s ) µ(x f ) 1,0 1,0 g(x) = 0 Fuzzy failure region X f : g(x) # 0 0,0 Fuzzy survival region X s : g(x) > 0 0,0 Fuzzy function Slide 5 of 45

6 Fuzzy first order reliability method Transformation of fuzzy random variables Space of the arbitrarily distributed random variable F(x) 1,0 µ = 1 µ = 0 Standard normal space Φ NN (y) 1,0 0,0 y = Φ x i 1 NN ( F(x) ) x µ(x) 1,0 0,0 y i,α = 0 l y i y i,α = 1 y i,α = 0 r y y Slide 6 of 45

7 Standard normal space complete fuzziness in h(y) = 0 Y fuzzy design point y B y 2 n NN (y) µ = 0 µ = 1 0 ψ 2 ψ 1 y 1 fuzzy survival region h(y) > 0 β fuzzy design point y B fuzzy reliability index β µ(β) 1.0 h(y) = 0 fuzzy failure region h(y) < 0 α β α l β α β α r β Slide 7 of 45

8 FFORM numerical realization Fuzzy input parameter = X 1 6 x = (...; x i ;...) β 0 X 1,α FORM x 2 0 X 2,α algorithm Fuzzy input parameter x 2 = X 2,α r X 1,α,α l µ( ) µ(x 2 ) 1,0 α 0,0 α-level optimization 0,0 α 1,0 x 2,α r X 2,α x 2,α l x 2 µ(β) 1,0 α Fuzzy reliability index β 0,0 β α l β α r Slide 8 of 45

9 Fuzzy first order reliability method Safety verification: β $ erf_β µ(β) fulfilled µ(β) not fulfilled µ(β) partially fulfilled 1,0 β 1,0 β µ 1 = 1,0 β µ 2 β 1 β 2 0,0 erf_β β α = 0 l β 0,0 β α = 0 r erf_β β 0,0 β α = 0 l erf_β β Slide 9 of 45

10 Example: Fuzzy stochatic safety assessment Slide 10 of 45

11 Example: Fuzzy stochastic safety assessment Reinforced Concrete Frame 6.00 m ν P V 0 ν p 0 ν P V 0 concrete C 20/25 reinforcement steel BSt i 16 2 i 16 4 i 16 cross section 500/ P H (1) 3 (2) 4 ν loadfactor P H = 10 kn P V 0 = 100 kn p 0 = 10 kn/m (3) k n 1 2 k n Loading Process dead load horizontal load P H vertical loads ν P V 0 and ν p V 0 fuzzy stochastic basic variable Slide 11 of 45

12 Features of the deterministic fundamental solution plane structural model with imperfect straight bars and layered cross sections numerical integration of the system of 1st order differential equations for the bars interaction of internal forces incremental-iterative solution technique under consideration of complex loading processes consideration of all essential geometrical and physical nonlinearities - large displacements and moderate rotations - realistic material description of reinforced concrete including cyclic and damage effects Slide 12 of 45

13 FFORM - Analysis I fuzzy probabilistic basic variables load factor ν ( ) extreme value distribution ex-max-type I (GUMBEL) m = < 5.7; 5.9; 6.0 > σ = < 8; 0.11; 0.12 > rotational spring stiffness k n (x 2 ) logarithmic normal distribution x 0,2 m σ x 2 x 2 = 0 MNm/rad = < 8.5; 9.0; 1 > MNm/rad = < 1.00; 1.35; 1.50 > MNm/rad f 1 ( ) 4.0 µ = 0 µ = 1 f 2 (x 2 ) 0.3 µ = 0 µ = x 2 [MNm/rad] Slide 13 of 45

14 FFORM - Analysis I (cont d) original space of the fuzzy probabilistic basic variables fuzzy joint probability density function, crisp limit state surface and fuzzy design point x 2 [MNm/rad] µ = 1.0 µ = 0.5 µ = g( ; x 2 ) > 0 f( ; x 2 ) s µ(x B ) s α=0, r s α=1 1.0 s α=0, l x B g( ; x 2 ) < 0 g( ; x 2 ) = 0 (from a geometrically und physically nonlinear structural analysis) Slide 14 of 45

15 FFORM - Analysis I (cont d) standard normal space crisp standard joint probability density function, fuzzy limit state surface and fuzzy design point y n NN (y 1 ; y 2 ) ψ 2 2 ψ 1 4 h(y 1 ; y 2 ) = 0 8 y 1 fuzzy reliability index, safety verification µ(β) β > 3.8 = req_β -2 β h(y 1 ; y 2 ) > 0-4 y B h(y 1 ; y 2 ) < µ = µ = 0.5 µ = β Slide 15 of 45

16 FFORM - Analysis II data and model uncertainty f 1 ( ) fuzzy probabilistic basic variable - load factor ν 4.0 extreme value distribution ex-max-type I (GUMBEL) m = < 5.7; 5.9; 6.0 > σ = < 8; 0.11; 0.12 > µ = 0 µ = 1 fuzzy model parameter rotational spring stiffness k n fuzzy triangular number k n µ(k n ) 1.0 = < 5; 9; 13 > MNm/rad k n [MNm/rad] Slide 16 of 45

17 FFORM - Analysis II (cont d) original space of the fuzzy probabilistic basic variables fuzzy probability density function, fuzzy limit state f 1 ( ) g( ) = ν - > 0 survival f 1 ( ) g(x 1 ) = ν - = 0 (from a geometically and physically nonlinear fuzzy structural analysis) g( ) = ν - < 0 failure µ(ν) µ = 0 µ = Slide 17 of 45

18 FFORM - Analysis II (cont d) fuzzy reliability index, safety verification µ(β) req_β = 3.8 β β µ(β) β 1 $ 3.8 = req_β µ(β) β 2 < 3.8 = req_β µ 1 = 1.0 β µ 2 = β β β Slide 18 of 45

19 Comparison: Analysis I - Analysis II analysis I: fuzzy reliability index β I analysis II: fuzzy reliability index β II µ(β Ι ) 1.0 µ(β ΙΙ ) β Ι β ΙΙ modified SHANNON&s entropy H u ( β) = k β=β α= 0, r β=β α= 0, l [ µ ( β) ln( µ ( β) ) + ( 1 µ ( β) ) ln( 1 µ ( β) )] dβ H u (β Ι ) = 1.41Ak < 2.48Ak = H u (β ΙΙ ) Slide 19 of 45

20 Influence - deterministic fundamental solution load factor ν: extreme value distribution ex-max-type I (GUMBEL) m = 5.9 x σ 1 = 0.11 rotational spring stiffness k n : beta distribution x min, 2 = 0 MNm/rad x max, x 2 2 = 12 MNm/rad m x 2 = 9.0 MNm/rad σ = 1.35 MNm/rad m x 2 = 9 8 computational model 4 0 x 2 [MNm/rad] g( ; x 2 ) = 0 0 m = geometrically and physically nonlinear only physically nonlinear only geometrically nonlinear β = β = β > 10 Slide 20 of 45

21 Thank you! Slide 21 of 45