Structural Design under Fuzzy Randomness

NSF workshop on Reliable Engineering Computing, September 15-17, 2004 Center for Reliable Engineering Computing, Georgia Tech Savannah, GA Structural Design under Fuzzy Randomness Michael Beer 1) Martin Liebscher 2) Bernd Möller 2) 1) Department of Mechanical Engineering and Materials Science Rice University, Houston, TX 2) Institute for Statics and Dynamics of Structures Dresden University of Technology, Dresden, Germany

Program 1 Introduction 2 Uncertainty models 3 Quantification of uncertainty 4 Fuzzy structural analysis 5 Fuzzy probabilistic safety assessment 6 Structural design based on clustering 7 Examples 8 Conclusions

Introduction (1) 1 Introduction the engineer s endeavor realistic structural analysis and safety assessment prerequisites: suitably matched computational models geometrically and physically nonlinear algorithms for numerical simulation of structural behavior appropriate description of structural parameters uncertainty has to be accounted for in its natural form

Introduction (2) 1 Introduction modeling structural parameters geometry? material? loading? foundation?

Introduction (3) 1 Introduction deterministic variable random variable x f(x) F(x) x! very few information! changing reproduction conditions! uncertain measured values! linguistic assessments! experience, expert knowledge alternative uncertainty models fuzzy variable fuzzy random variable

Fuzzy sets 2 Uncertainty models α-level set α-discretization X α = { x X µ X(x) α } X {( X ( ))} = X α ; µ α µ X (x) X α i α X αi, αk i α k αi, membership function α α k k (0;1] α k x αk l X αk x αk r x support S(X)

Fuzzy random variables (1) 2 Uncertainty models fuzzy random variable X fuzzy result of the uncertain mapping Ω 6 F(ú n ) Ω... space of the random elementary events ω F(ú n )... set of all fuzzy numbers x in ú n X is the fuzzy set of all possible originals X j original X j µ(x) ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 x 1 x 4 ω 0 Ω x 2 x 6 x 3 x 5 real valued random variable X that is completely enclosed in X x 0 X = ú 1

Fuzzy random variables (2) 2 Uncertainty models fuzzy probability distribution function F(x) F(x) = bunch of the F j (x) of the originals X j of X F(x) = {( F (x); (F (x)))} α µ F (x) = [ F (x); F (x)] α µ (F F α α l α α r (x)) = α α (0;1] 1 max P X j α l (x) = j, α ; = t µ(f(x)) x, t X t k F(x i ) F α=0 r (x i ) F α=0 l (x i ) x = k ú n ;1 k x, t X = Fα r (x) = max P X j, α = t j tk < xk ;k = 1,..., n ú n F(x) F α=1 (x i ) n 0.5 f(x) f(x) F(x) x i µ = 0 µ = 1 x µ = 0 µ = 1 x

Fuzzy random variables (3) 2 Uncertainty models special case fuzzy random variable X special case real random variable X µ(f(x)) F(x) 1 1 0 µ(f(x)) 1 1 F(x) fuzzy variable x µ(f(x)) F(x) 1 1 1 0 x 0 F(x) 1 x µ(x) 1 x 0 x 0 x

3 Quantification of uncertainty Quantification of fuzzy variables (fuzzification) objective and subjective information, expert knowledge limited data µ(β D ) n 4 β D = <7.5; 14.17; 57.5> linguistic assessment µ(β D ) low medium high 0.5 2 1.33 0.5 0 0 10 30 50 β D [N/mm²] single uncertain measurement µ(d) 0 10 30 50 β D [N/mm²] expert experience µ(x) crisp kernel set fuzzy interval d x

Quantification of fuzzy random variables 3 Quantification of uncertainty statistical information comprising partially non-stochastic uncertainty small sample size fuzzification of the uncertainty of statistical estimations and tests non-constant reproduction conditions that are known in detail separation of fuzziness and randomness by constructing groups non-constant, unknown reproduction conditions; uncertain measure values fuzzification of the sample elements followed by statistical evaluation s α=0 r s s α=1 s α=0 l _ x α=0 l F(x) with fuzzy parameters and fuzzy functional type _ x α=1 _ x _ x α=0 r F(x) µ = 1 µ = 0, interaction without interaction F(x) x

Basic numerical solution procedure 4 Fuzzy structural analysis fuzzy input variable x 1 = X 1 x 1 x = (...; x i ;...) 6 z = (...; z j ;...) x 1 0 X 1,α mapping x 2 0 X 2,α operator fuzzy input variable x 2 = X 2 x 1,α r X 1,α x 1,α l µ(x 1 ) µ(x 2 ) z j 0 Z j,α α α-level optimization α x 2,α r X 2,α x 2,α l x 2 µ(z j ) α fuzzy result variable z j = Z j z j,α l Z j,α z j,α r z j

Modified evolution strategy 4 Fuzzy structural analysis x 2 crisp input subspace X α x 2,α r lower distance bound upper distance bound for generating offspring points starting point improvement of z j no improvement of z j optimum point x opt,α 7 2 6 x 2,α l 5 4 0 1 3 x 1,α l x 1,α r x 1 post-computation recheck of all results for optimality

Fuzzy First Order Reliability Method (1) 5 Fuzzy probabilistic safety assessment original space of the basic variables fuzzy randomness Y f(x) fuzziness Y g(x) = 0 x 2 joint fuzzy probability density function f(x) µ = 1 µ = 0 fuzzy limit state surface g(x) = 0 fuzzy survival region X s : g(x) > 0 µ = 0 µ = 1 fuzzy failure region X f : g(x) < 0 x 1

5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (1) 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 ) g(x) = 0 fuzzy failure region X f : g(x) # 0 fuzzy survival region X s : g(x) > 0 x 1 x 1

Fuzzy First Order Reliability Method (1) 5 Fuzzy probabilistic safety assessment original space of the basic variables fuzzy randomness Y f(x) fuzziness Y g(x) = 0 fuzzy design point x B P f = x... f (x) X ; x f X s dx x 2 joint fuzzy probability density function f(x) µ = 1 fuzzy µ = 0 limit state µ(x surface g(x) B ) = 0 1 s Fuzzy First Order Reliability Method fuzzy survival region X s : g(x) > 0 µ = 0 µ = 1 x B t fuzzy failure region X f : g(x) < 0 x 1

5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (2) complete fuzziness in h(y) = 0 y 2 n NN (y) h(y) = 0 0 ψ 2 ψ 1 standard normal space fuzzy design point y B fuzzy reliability index β y 1 µ(β) { p ( X ) ; m } β t i j fuzzy survival region h(y) > 0 β fuzzy design point y B µ = 0 µ = 1 α β α fuzzy failure region h(y) < 0 β α l β α r β

5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (3) selection of an α-level numerical solution B α-level optimization determination of an original of each basic variable determination of one trajectory of f(x) determination of a single point in the space of the fuzzy structural parameters determination of one element of g(x) = 0 computation of one element β i of β α µ(β) β β α β i comparison of β i with β α l and β α r α number of α-levels sufficient? β β α l β α r

6 Structural design based on clustering Structural design based on clustering B concept fuzzy structural analysis and fuzzy probabilistic safety assessment set M(x) of discrete points x i in the input subspace, set M(z) of the assigned discrete points z i in the result subspace design constraints CT h subdividing M(z) into permissible and non-permissible points assigned permissible and non-permissible points within M(x) separate clustering of the permissible and non-permissible points from M(x) elimination of all non-permissible clusters including intersections with permissible clusters permissible clusters of design parameters representing alternative design variants

Design constraint µ(x 1, x 2 ) 1 z = f(x 1, x 2 ) 6 Structural design based on clustering µ(z) 1 α α 0 x 2 α 0 fuzzy result variable z z fuzzy input variables x 1 and x 2 x 1 points from α-level optimization permissible points non-permissible points permissible structural design variants, assessment of alternatives

Cluster analysis 6 Structural design based on clustering k-medoid method fuzzy cluster method x 2 crisp C1 C2 x 2 C1 C2 " c $ 0.25 " c $ C2 representative objects dissimilarity C3 similarity C3 C1 C3 determination of representative objects assignment of remaining objects to the most similar representative objects x 1 minimization of the functional c' k jv'1 j x 1 n i,j'1 µr iv µr jv d(i,j) 2 j n j'1 µr jv

6 Structural design based on clustering Design variants B modified fuzzy input variables modeling selection of a cluster or cluster combination definition of the support of the fuzzy input variable based on the cluster boundaries definition of the mean value x with :(x) = 1 construction of the membership function oriented to the primary fuzzy input variable based on " c -levels from fuzzy clustering representative object or prototype as mean value geometrical cluster center as mean value :(x) 1 0.5 prototype " c = 0.5 " c = 0.25 x

6 Structural design based on clustering Assessment of alternative design variants (1) defuzzification JAIN-algorithm :(z) 1 k=2 k=1 CHEN-algorithm :(z) 1 : s (z) k=1 : s (z) k=0.5 maximizing set k z inf[ z] µ(z max ) z z = sup[ z] inf [ z] z z z z s z l [ ] µ (z) = sup min µ(z);µ(z ) z z z 0 z z r max z 0 = µ s zr zl + zl with: max z z l z 0 z r minimizing set z min with: k sup[ z] z µ(z min ) z z = sup[ z] inf [ z] z z z z additional methods: level rank method centroid method z

6 Structural design based on clustering Assessment of alternative design variants (2) constraint distance :(z) robustness modified SHANNON s entropy [ ] H(z) = k µ(z) ln(µ(z)) + (1 µ ( z)) ln(1 µ(z)) dz z z z 0 d s perm_z z relative sensitivity measure d s max uncertainty of the fuzzy result variables in relation to the uncertainty of the fuzzy input variables B(z ) j = H(z j) H(x ) i j min

Deterministic computational model 7 Examples geometrically and physically nonlinear numerical analysis of plane (prestressed) reinforced concrete bar structures imperfect straight bars with layered cross sections numerical integration of a system of 1st order ODE for the bars interaction between internal forces incremental-iterative solution technique to take account 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

Reinforced concrete frame 7 Examples 6000 mm νap V 0 νap 0 νap V 0 cross sections: 500/350 concrete: C 20/25 reinforcement steel: BSt 420 4 i 16 2 i 16 4 i 16 stirrups: i 8 s = 200 concrete cover: c = 50 8000 P H (1) k n 3 (2) 4 ν load factor P H = 10 kn P V 0 = 100 kn p 0 = 10 kn/m 1 2 (3) k n loading process: 1) dead load 2) horizontal load P H 3) vertical loads νap V 0 and νap 0

Fuzzy structural analysis (1) 7 Examples fuzziness fuzzy rotational spring stiffness k n = < 5; 9; 13 > MNm/rad fuzzy load factor ν = < 5.5; 5.9; 6.7 > µ(k n ) deterministic values µ(ν) 5 9 13 k n [MNm/rad] 5.5 5.9 6.7 ν

Fuzzy structural analysis (2) 7 Examples fuzzy load-displacement dependency (left-hand frame corner, horizontal) ν 8.0 6.7 5.9 5.5 4.0 2.0 1.57 2.16 fuzzy structural response µ = µ = 4.0 6.0 8.0 v H (3) [cm] fuzzy displacement µ(v H (3)) 0 0.75 0.50 0.25 0.144 deterministic solution 0 1.57 2.63 4.06 7.21 10.69 2.16 v H (3) [cm]

Fuzzy structural analysis (3) 7 Examples fuzzy failure load influence of the deterministic fundamental solution nonlinearities considered: g + p = geometrical and physical p = only physical g = only geometrical µ(ν) 0.5 ν g+p ν p ν g 6.43 7.24 7.63 8.11 ν 27.15 32.70 37.60 ν

FFORM B analysis I (1) 7 Examples load factor ν: X 1 extreme value distribution Ex-Max Type I (GUMBEL) m = < 5.7; 5.9; 6.0 > σ = < 8; 0.11; 0.12 > x 1 x 1 fuzzy randomness 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 = < 0; 1.35; 1.50 > MNm/rad f(x 1 ) 4.0 µ = 0 µ = 1 f(x 2 ) 0.3 µ = 0 µ = 1 2.0 0.2 0.1 5.7 5.9 6.0 x 1 8.5 1 9.0 x 2 [MNm/rad]

FFORM B analysis I (2) 7 Examples original space of the fuzzy probabilistic basic variables joint fuzzy probability density function crisp limit state surface fuzzy design point x 2 [MNm/rad] 12 8 4 0 0 µ = µ = 0.5 µ = g(x 1 ; x 2 ) > 0 f(x 1 ; x 2 ) s µ(x B ) s α=0 r s α=1 5 6 s α=0 l x B g(x 1 ; x 2 ) < 0 7 g(x 1 ; x 2 ) = 0 8 x 1

FFORM B analysis I (3) 7 Examples crisp standard joint probability density function fuzzy limit state surface fuzzy design point fuzzy reliability index y 2 h(y 1 ; y 2 ) = 0 safety verification 1-2 0-2 h(y 1 ; y 2 ) > 0-4 n NN (y 1 ; y 2 ) ψ 2 β 2 ψ 1 4 standard normal space y B 8 y 1 h(y 1 ; y 2 ) < 0 µ = µ = 0.5 µ = 0.5 µ(β) 3.851 4.662 4.255 β > 3.8 = req_β FORM-solution 5.609 6.684 β

FFORM B analysis II (1) 7 Examples fuzziness and fuzzy randomness f(x 1 ) fuzzy probabilistic basic variable X 1 : load factor ν 4.0 extreme value distribution Ex-Max Type I (GUMBEL) m x = < 5.7; 5.9; 6.0 > σ 1 = < 8; 0.11; 0.12 > x 1 µ = 0 µ = 1 fuzzy structural parameter: rotational spring stiffness k n fuzzy triangular number k n µ(k n ) = < 5; 9; 13 > MNm/rad 2.0 5.7 5.9 6.0 x 1 5 9 13 k n [MNm/rad]

FFORM B analysis II (2) 7 Examples original space of the fuzzy probabilistic basic variables fuzzy probability density function fuzzy limit state surface f(x 1 ) 4.0 2.0 g(x 1 ) = ν -x 1 > 0 survival f(x 1 ) µ = 0 µ = 1 g(x 1 ) = ν -x 1 = 0 ν g+p (from fuzzy structural analysis) µ(ν) g(x 1 ) = ν -x 1 < 0 failure fuzzy integration limit 0.5 5.7 5.9 6.0 6.43 7.24 7.63 x 1

FFORM B analysis II (3) 7 Examples fuzzy reliability index, safety verification µ(β) req_β = 3.8 β II FORM-solution β 2 < 3.8 = erf_β 0.4 β 2 2.534 3.935 β 1 5.214 6.592 7.557 β comparison with the result from FFORM-analysis I β 1 $ 3.8 = req_β µ(β) β I modified SHANNON&s entropy H u (β Ι ) = 1.41Ak < 2.48Ak= H u (β ΙΙ ) 3.851 4.662 6.684 β

Structural design based on clustering (1) 7 Examples load factor ν: X 1 extreme value distribution Ex-Max Type I (GUMBEL) m = < 5.7; 5.9; 6.0 > σ = < 8; 0.11; 0.12 > x 1 x 1 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 = < 0; 1.35; 1.50 > MNm/rad f(x 1 ) m = 0 m = 1 f(x 2 ) m = 0 m = 1 5.7 5.9 6.3 x 1 8.5 1 9.0 x 2 [MNm/rad]

Structural design based on clustering (2) 7 Examples fuzzy reliability index design constraint µ($) 0.4 req_$ = 3.8 reliability index req_$ 3.8 2.49 4.7 6.69 $ fuzzy cluster analysis modified fuzzy probabilistic basic variables design variant 1 2 load factor ν rotational spring stiffness k n [MNm/rad] m - σ - x 1 σ - x 1 m - x 2 x 2 < 5.85; 5.90; 6.0 > < 8; 0.10; 0.12 > < 8.5; 9.0; 9.6 > < 1.30; 1.40; 1.50> < 5.70; 5.85; 6.0 > < 8; 0.10; 0.11 > < 8.9; 9.5; 1 > < 2; 1.14; 1.26 >

Structural design based on clustering (3) 7 Examples FFORM-analysis with the modified fuzzy probabilistic basic variables design variant 1 µ(ß 1 ) design variant 2 µ(ß 2 ) 3.80 5.04 5.87 4.452 5.4 6.676 ß 1 ß 2 defuzzification after CHEN relative sensitivity z 01 = 4.60 z 02 = 5.12 B 1 = 92.67 B 2 = 77.48

Conclusions 8 Conclusions consideration of non-stochastic uncertainty in structural analysis, design, and safety assessment enhanced quality of prognoses regarding structural behavior and safety direct design of structures by means of nonlinear analysis

Advertisement A detailed explanation of all concepts presented and much more may be found in the book: Bernd Möller Michael Beer Fuzzy Randomness Uncertainty Models in Civil Engineering and Computational Mechanics Springer-Verlag, 2004 Dept. of Mechanical Engineering and Materials Science, Rice University, Houston, TX