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

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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 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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Introduction (1) 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 1 Introduction the engineer‘s endeavor Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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modeling structural parameters geometry ? material ? loading ? foundation ? Introduction (2) 1 Introduction Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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deterministic variable random variable x f(x) x F(x) very few information ! changing reproduction conditions ! uncertain measured values ! linguistic assessments ! experience, expert knowledge ! alternative uncertainty models fuzzy variable fuzzy random variable Introduction (3) 1 Introduction very few information ! changing reproduction conditions ! uncertain measured values ! linguistic assessments ! experience, expert knowledge ! Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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X kk X (x) 1.0 0.0 kk Fuzzy sets -level set support S(X) ~ -discretization x k l x k r x 2 Uncertainty models membership function Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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space of the random elementary events ~ F( n ) set of all fuzzy numbers x in n fuzzy random variable X ~ 11 22 33 44 55 66 (x) 1.0 0.0 x X = 1 x6x6 ~ x5x5 ~ x4x4 ~ x3x3 ~ x2x2 ~ x1x1 ~ Fuzzy random variables (1) real valued random variable X that is completely enclosed in X ~ original X j X is the fuzzy set of all possible originals X j ~ fuzzy result of the uncertain mapping F( n ) ~ 2 Uncertainty models Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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xixi 1.00.0 (F(x)) F(x i ) ~ F(x) 1.0 0.0 0.5 F(x) ~ x = 0 = 1 f(x) ~ x = 0 = 1 fuzzy probability distribution function F(x) ~ F =0 l (x i ) F =0 r (x i ) F =1 (x i ) únún únún F(x) = bunch of the F j (x) of the originals X j of X ~ ~ Fuzzy random variables (2) 2 Uncertainty models Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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Fuzzy random variables (3) fuzzy random variable X ~ (F(x)) F(x) x 0 1 1 special case real random variable X (F(x)) F(x) x 0 1 1 x 0 1 fuzzy variable x ~ (F(x)) F(x) x 0 1 1 (x) x 0 1 special case 2 Uncertainty models Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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(D)(D) 1.0 0.5 0.0 D = ~ Quantification of fuzzy variables (fuzzification) limited data objective and subjective information, expert knowledge 010 3050 D [N/mm²] n 4 2 1.33 0 linguistic assessment low mediumhigh (D)(D) 1.0 0.5 0.0 010 3050 D [N/mm²] single uncertain measurement (d) 1.0 0.0 d expert experience crisp kernel set fuzzy interval x (x) 1.0 0.0 3 Quantification of uncertainty Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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s =0 l s =0 r x =0 r _ x =0 l _ F(x) ~ = 0, interaction s x _ x F(x) 1.0 0.0 = 1 x =1 _ s =1 F(x) with fuzzy parameters and fuzzy functional type ~ without interaction Quantification of fuzzy random variables 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 3 Quantification of uncertainty Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzzy result variable z j = Z j ~ z j Z j, 0.0 1.0 zjzj Z j, z j, l z j, r (z j ) ~ mapping operator x 1 X 1, x 2 X 2, x = (...; x i ;...) z = (...; z j ;...) ~~~ ~ Basic numerical solution procedure 0.01.0 0.0 1.0 (x 1 ) (x 2 ) fuzzy input variable x 2 = X 2 ~ fuzzy input variable x 1 = X 1 ~ X 1, x 1, l x 1, r x1x1 X 2, x 2, l x 2, r x2x2 -level optimization ~ ~ 4 Fuzzy structural analysis Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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improvement of z j starting point lower distance bound upper distance bound for generating offspring points Modified evolution strategy x 1, l x 1, r x1x1 x 2, l x 2, r x2x2 no improvement of z j 1 0 2 4 Fuzzy structural analysis crisp input subspace X Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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improvement of z j starting point lower distance bound upper distance bound for generating offspring points Modified evolution strategy x 1, l x 1, r x1x1 x 2, l x 2, r x2x2 no improvement of z j 1 0 2 3 4 5 6 7 optimum point x opt, 4 Fuzzy structural analysis crisp input subspace X post-computation recheck of all results for optimality Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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survival region limit state surface g(x) = 0 failure region Fuzzy First Order Reliability Method (1) x2x2 original space of the basic variables x1x1 X s : g(x) > 0 X f : g(x) < 0 5 Fuzzy probabilistic safety assessment joint probability density function f(x) fuzzy fuzzy randomness f(x) ~ = 1 = 0 ~ ~ fuzziness g(x) = 0 fuzzy ~ ~ ~ ~ ~ = 1 = 0 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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g(x) =1 0 x2x2 x1x1 (xf)(xf) 1.0 0.0 g(x) =1 = 0 g(x) = 0 ~ fuzzy failure region X f : g(x) 0 ~ ~ 5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (1) g(x) =1 > 0 x2x2 x1x1 (xs)(xs) 1.0 0.0 g(x) =1 = 0 fuzzy survival region X s : g(x) > 0 ~ ~ Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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survival region limit state surface g(x) = 0 failure region Fuzzy First Order Reliability Method (1) x2x2 original space of the basic variables x1x1 X s : g(x) > 0 X f : g(x) < 0 5 Fuzzy probabilistic safety assessment joint probability density function f(x) fuzzy fuzzy randomness f(x) ~ = 1 = 0 ~ ~ fuzziness g(x) = 0 fuzzy ~ ~ ~ ~ ~ = 1 = 0 Fuzzy First Order Reliability Method ~ ~ fuzzy design point x B 1 s t xBxB ~ (xB)(xB) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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1.0 0.0 l r ~ ~ fuzzy design point y B ~ 22 11 ~ ~ y2y2 h(y) = 0 ~ h(y) < 0 ~ fuzzy failure region = 1 = 0 NN (y) y1y1 ~ complete fuzziness in h(y) = 0 standard normal space 0 5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (2) fuzzy reliability index ~ ~ fuzzy design point y B h(y) > 0 ~ fuzzy survival region Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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1.0 0.0 ii l r numerical solution B -level optimization selection of an -level computation of one element i of comparison of i with l and r number of -levels sufficient ? determination of one trajectory of f(x) determination of an original of each basic variable ~ determination of a single point in the space of the fuzzy structural parameters determination of one element of g(x) = 0 ~ 5 Fuzzy probabilistic safety assessment Fuzzy First Order Reliability Method (3) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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Structural design based on clustering B concept fuzzy structural analysis and fuzzy probabilistic safety assessment 6 Structural design based on clustering 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 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzzy input variables x 1 and x 2 ~ z = f(x 1, x 2 ) ~ points from -level optimization permissible points non-permissible points Design constraint permissible structural design variants, assessment of alternatives 6 Structural design based on clustering (x 1, x 2 ) 1 0 x1x1 x2x2 ~ (z) fuzzy result variable z 1 0 z Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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C3 C2 C1 dissimilarity similarity assignment of remaining objects to the most similar representative objects x1x1 x2x2 k-medoid method crisp C3 C2 C1 c 0.0 C1 C2 C3 c 0.25 fuzzy cluster method x1x1 x2x2 minimization of the functional Cluster analysis 6 Structural design based on clustering Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX determination of representative objects representative objects

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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 modeling (x) x 1 0.5 0.0 c = 0.25 c = 0.5 prototype construction of the membership function representative object or prototype as mean value based on c -levels from fuzzy clustering geometrical cluster center as mean value oriented to the primary fuzzy input variable Design variants B modified fuzzy input variables 6 Structural design based on clustering Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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JAIN-algorithm z (z) 0.0 1 additional methods: level rank method centroid method s (z) k=2 k=1 k=0.5 maximizing set with: z0z0 zlzl zrzr Assessment of alternative design variants (1) defuzzification 6 Structural design based on clustering s (z) CHEN-algorithm z (z) 0.0 1 k=1 minimizing set with: z0z0 zlzl zrzr Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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constraint distance perm_z (z) z z0z0 dsds dsds max relative sensitivity measure uncertainty of the fuzzy result variables in relation to the uncertainty of the fuzzy input variables min robustness modified SHANNON‘s entropy Assessment of alternative design variants (2) 6 Structural design based on clustering Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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Deterministic computational model imperfect straight bars with layered cross sections geometrically and physically nonlinear numerical analysis of plane (prestressed) reinforced concrete bar structures 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 7 Examples Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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k (3)(1) (2) 43 21 k PHPH P H =10 kN PHPH 2)horizontal load P H Reinforced concrete frame cross sections: 500/350 concrete: C 20/25 reinforcement steel: BSt 420 4 16 2 16 stirrups: 8 s = 200 concrete cover: c = 50 6000 mm 8000 loading process:1)dead load 7 Examples load factor A P V 0 A p 0 P V 0 = 100 kN p 0 =10 kN/m 3)vertical loads A P V 0 and A p 0 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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Fuzzy structural analysis (1) fuzziness fuzzy rotational spring stiffness k = MNm/rad ~ (k ) 1.0 0.0 k [MNm/rad] 1395 fuzzy load factor = ~ ( ) 1.0 0.0 6.75.95.5 deterministic values 7 Examples Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzzy displacement (v H (3)) 1.00 0.00 0.144 0.25 0.50 0.75 v H (3) [cm] 2.631.570.04.067.2110.69 2.16 deterministic solution fuzzy structural response fuzzy load-displacement dependency (left-hand frame corner, horizontal) 8.0 0.0 v H (3) [cm]2.161.570.0 6.7 5.9 5.5 4.0 2.0 4.06.08.0 = 0.0 = 1.0 Fuzzy structural analysis (2) 7 Examples Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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g ~ 32.7027.1537.60 g = only geometrical 1.0 0.0 7.246.43 0.5 7.63 fuzzy failure load g+p ~ nonlinearities considered: g + p = geometrical and physical influence of the deterministic fundamental solution Fuzzy structural analysis (3) 7 Examples 8.11 p p = only physical Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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FFORM B analysis I (1) load factor : X 1 extreme value distribution Ex-Max Type I (GUMBEL) m= = fuzzy randomness x1x1 x1x1 ~ ~ f(x 1 ) 4.0 2.0 x1x1 5.75.96.0 = 0 = 1 0.0 ~ rotational spring stiffness k : X 2 logarithmic normal distribution x 0,2 = 0 MNm/rad m= MNm/rad = MNm/rad x2x2 x2x2 ~ ~ = 0 = 1 f(x 2 ) 0.3 0.2 x 2 [MNm/rad] 8.5 9.0 10.0 0.0 0.1 ~ 7 Examples Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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g(x 1 ; x 2 ) = 0 g(x 1 ; x 2 ) > 0 crisp limit state surface g(x 1 ; x 2 ) < 0 xBxB ~ s =0 l s =0 r s =1 s 0.0 1.0 (xB)(xB) fuzzy design point joint fuzzy probability density function original space of the fuzzy probabilistic basic variables x 2 [MNm/rad] f(x 1 ; x 2 ) ~ 0 4 8 12 x1x1 87650 = 1.0 = 0.5 = 0.0 7 Examples FFORM B analysis I (2) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzzy reliability index ()() 1.0 0.5 0.0 6.684 5.6094.662 4.255 3.851 FORM-solution 4.662 y2y2 y1y1 842-2 0 1 -4 NN (y 1 ; y 2 ) h(y 1 ; y 2 ) > 0 ~ h(y 1 ; y 2 ) = 0 ~ h(y 1 ; y 2 ) < 0 ~ = 0.0 = 0.5 = 1.0 fuzzy limit state surface crisp standard joint probability density function standard normal space > 3.8 = req_ ~ safety verification 7 Examples FFORM B analysis I (3) yByB ~ ~ 11 ~ 22 ~ fuzzy design point Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzziness and fuzzy randomness fuzzy structural parameter: rotational spring stiffness k fuzzy triangular number k = MNm/rad ~ ~ (k ) 1.0 0.0 k [MNm/rad] 1395 fuzzy probabilistic basic variable X 1 : load factor extreme value distribution Ex-Max Type I (GUMBEL) m= = x1x1 x1x1 ~ ~ f(x 1 ) 4.0 2.0 x1x1 5.75.96.0 = 0 = 1 0.0 ~ 7 Examples FFORM B analysis II (1) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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fuzzy probability density function f(x 1 ) 4.0 2.0 5.75.96.0 0.0 x1x1 f(x 1 ) ~ = 0 = 1 original space of the fuzzy probabilistic basic variables fuzzy integration limit g(x 1 ) = - x 1 = 0 ~~ g(x 1 ) = - x 1 > 0 survival ~~ g(x 1 ) = - x 1 < 0 failure ~~ ( ) 1.0 0.5 0.0 6.437.247.63 fuzzy limit state surface g+p (from fuzzy structural analysis) ~ 7 Examples FFORM B analysis II (2) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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3.8 = req_ ~ < 3.8 = erf_ ~ 11 ~ 22 ~ FORM-solution 5.214 6.592 5.2147.557 fuzzy reliability index, safety verification ()() 1.0 0.4 0.0 3.9352.534 req_ = 3.8 II ~ comparison with the result from FFORM-analysis I ()() 1.0 0.0 6.684 4.6623.851 II ~ H u ( ) = 1.41 A k < 2.48 A k = H u ( ) ~~ modified SHANNON & s entropy 7 Examples FFORM B analysis II (3) Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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f(x 1 ) x1x1 5.75.96.3 m = 0 m = 1 m = 0 m = 1 f(x 2 ) x 2 [MNm/rad] 8.5 9.0 10.0 rotational spring stiffness k : X 2 logarithmic normal distribution x 0,2 = 0 MNm/rad m= MNm/rad = MNm/rad x2x2 x2x2 ~ ~ ~ load factor : X 1 extreme value distribution Ex-Max Type I (GUMBEL) m= = x1x1 x1x1 ~ ~ ~ Structural design based on clustering (1) 7 Examples Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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req_ = 3.8 reliability index design constraint req_ ≥ 3.8 4.76.69 ()() 1.0 0.4 0.0 2.49 fuzzy reliability index Structural design based on clustering (2) 7 Examples 1 2 fuzzy cluster analysis modified fuzzy probabilistic basic variables design variant Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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FFORM-analysis with the modified fuzzy probabilistic basic variables defuzzification after CHEN relative sensitivity z 01 = 4.60z 02 = 5.12 B 1 = 92.67B 2 = 77.48 design variant 1 Structural design based on clustering (3) 7 Examples design variant 2 Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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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 8 Conclusions Conclusions Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX

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

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