1. Institute for Structural Analysis Uncertain processes and numerical monitoring of structures Wolfgang Graf Bernd Möller Matthias Bartzsch www.tu-dresden.de/isd www.uncertainty-in-engineering.net 2008 February 20 GeorgiaTech Savannah

2. Institute for Structural Analysis 2 REC 2008 1Motivation 2Modification of structures 3Fuzzy functions 4Fuzzy structural analysis 5Examples Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

3. Institute for Structural Analysis 3 REC 2008 Modification of (composite) structures Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

4. Institute for Structural Analysis 4 REC 2008 Load and modification process Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples g 1 +p p g concrete g steel g2g2

5. Institute for Structural Analysis 5 REC 2008 Load and modification process reinforced column P with strengthening Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

6. Institute for Structural Analysis 6 REC 2008 material modification modification of structural members/ modification of support conditions modification of cross sections structural modification prestressing/strengthening ii  i+1  i+2 Modification of structures as process Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

7. Institute for Structural Analysis 7 REC 2008 time  system parameter construction phase exploitation phase uncertain process reconstruction phase exploitation phase system lifetime Time dependent modification of system parameters Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

8. Institute for Structural Analysis 8 REC 2008 z(t)z(t) vector of structural responses g(t)g(t) dead load p(t)p(t) statical and dynamical external loads FV(t)FV(t) prestressing forces (internal and external prestressing) T(t)T(t) parameters of temperature A(t), I(t), L(t) parameters of geometry (e.g., cross sections, system dimensions, location of reinforcement, and prestressing) E(t)E(t) material parameters t = ( θ, τ, φ ) spatial coordinates θ= θ 1, θ 2, θ 3, time τ, further parameters φ modification process:discontinuous and uncertain parameters z(t) = f g(t), p(t), F V (t), T(t), A(t), I(t), L(t), E(t) ( ) time dependent values in the modification process Mathematical modeling Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

9. Institute for Structural Analysis 9 REC 2008 uncertain input parameters uncertain/ crisp mapping model uncertain result parameters  spatial coordinates  time x( , , ) z ( , , ) ~~ time dependent; continuous/ discontinuous time dependent/ time independent time dependent; continuous/ discontinuous Modification as uncertain discontinuous process further parameters Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

10. Institute for Structural Analysis 10 REC 2008 deterministic data model x x fuzzy- randomness f(x) x  =0  =1 ~ f(x) ={(f(x);  f (f(x))) | f  f };  f (f(x))  0  f  f ~ fuzziness x ={(x;  x (x)) | x  X };  x (x)  0  x  X ~ x  (x) 1.0  (x) randomness x f(x) F(x) f(x), F(x) 1 Uncertain parameters of the modification process Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

11. Institute for Structural Analysis 11 REC 2008 μ X (x) 1.0 0.0 kk x xαklxαkl xαkrxαkr Xα kXα k support S(X) ~ X  = x  X │  x (x) ≥  {} X = X  ;  (X  ) {()} ~ X  i X  k  i,  k  i ≥  k  i ;  k  (0,1] α -level set α -discretization Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

12. Institute for Structural Analysis 12 REC 2008 x 5 ( τ ) = 0.9 sin(0.9 τ - 0.1) μ(x 5 ( τ )) = 0.0 x(t) = x(s, t) ~~ x 1 ( τ ) = sin( τ ) μ(x 1 ( τ )) = 1.0 x 3 ( τ ) = 1.05 sin(1.05 τ + 0.05) μ(x 3 ( τ )) = 0.5 x 2 ( τ ) = 0.95 sin(0.95 τ - 0.05) μ(x 2 ( τ )) = 0.5 x 4 ( τ ) = 1.1 sin(1.1 τ + 0.1) μ(x 4 ( τ )) = 0.0 x(  ) = x(s,  ) = s 1 · sin(s 2 ·  + s 3 ) ~~~~~ with s = (s 1, s 2, s 3 ) s 1 = ~~~~ ~ s 2 = ~ s 3 = ~ trajectory μ μμ 1.0 αkαk αkαk αkαk s2s2 s3s3 s1s1 crisp subspace S α with s  S α example 1,0 0,0 -1,0 246810  [rad] x Bunch parameter representation Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

13. Institute for Structural Analysis 13 REC 2008 saltus: lim x(, , φ ) = lim x(, , φ)  →  +  →  - μ x   1 1 ~~ x()x() ~ Discontinuity in fuzzy function Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

14. Institute for Structural Analysis 14 REC 2008 x(, , φ ) = (x t = x(, , φ))   |   F(T),  , φ | , φ  T { } ~~~~~~~  1 1 ~ μ x  x()x() ~~ Discontinuity with uncertain point of time Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

15. Institute for Structural Analysis 15 REC 2008 x(t): T F(X) ~ fundamental setset of fuzzy variables t = ( , , φ ) set of fuzzy variables x(t) = x t = x(t) t │t  T {} ~~~ Fuzzy function: x(  ) = x  = x(  )  │   T {} ~~~ fuzzy process: A system parameter x time   (x) 11 22 33 ~  2 = {(  ;  (  )) |   T} x(  ) ~ system modification increment Motivation Modification of strutures Fuzzy functions Fuzzy structural analysis Examples

16. Institute for Structural Analysis 16 REC 2008 x2x2   x1x1 x1x1 x2x2 ~ ~  x()x() ~ crisp time dependent fuzzy input subspace fuzzy input functions x i (  ) ~ time dependent fuzzy input space Time dependent fuzzy structural analysis (1) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

17. Institute for Structural Analysis 17 REC 2008 x2x2   x1x1 x1x1 x2x2 ~ ~  x()x() ~ crisp time dependent fuzzy input subspace Time dependent fuzzy structural analysis (2) time dependent fuzzy input space fuzzy input functions x i (  ) ~ Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

18. Institute for Structural Analysis 18 REC 2008 x2x2   x1x1 x1x1 x2x2 ~ ~  x()x() ~  z  1 0 z(  ) ~ 11 fuzzy result function deterministic fundamental solution Time dependent fuzzy structural analysis (3) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

19. Institute for Structural Analysis 19 REC 2008 Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples ~ 1  (p 1 ) p1p1 p1p1 ~ discrete fuzzy parameter 22 ii s2(2)s2(2) E0(1)E0(1) E 0 (  2 )  E 0 (  3 )  -level optimization 11 s1(1)s1(1) ~  fuzzy function x(s,θ,  ) with fuzzy functional values at  1,  2 mapping model M(  3 ) 33 z(θ,  3 ) ~ z(θ,  1 ) ~ fuzzy result values at particular points in time mapping model M(  1 ) 11  mapping model M(  2 ) 22 z(θ,  2 ) ~ input subspaces ~ Time dependent fuzzy structural analysis (4)

20. Institute for Structural Analysis 20 REC 2008 e.g. ADINA LS-DYNA … inhouse software: geometrically and physically nonlinear analysis of concrete, reinforced concrete, textile reinforced concrete, prestressed concrete, masonry and steel plane and spatial bar structures (STATRA) or folded plate structures (FALT-FEM) Deterministic fundamental solution (1) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

21. Institute for Structural Analysis 21 REC 2008 Deterministic fundamental solution (2) geometrically and physically nonlinear analysis of plane reinforced concrete, prestressed concrete, and steel bar structures incremental formulation of second order differential equation system = A(θ 1, z) (n-1) · Δz(θ 1 ) + Δb(θ 1, z) dΔz(θ1)dΔz(θ1) dθ1dθ1 (n) [k] (n) [k] (n) [k-1] + d(θ 1 ) (n-1) · Δz 1 (θ 1 ) + m(θ 1 ) (n-1) · Δz 1 (θ 1 ) (n) [k] (n) [k] ··· K T (n-1) · Δv (n) + D (n-1) · Δv (n) + M (n-1) · Δv (n) = ΔP (n) – ΔF (n) + ΔΔF (n-1) [k] ··· z (θ 1 ) = { v; S } = { u, v, φ, s; N, V, M, N s } (n)... increment [k]... iteration step Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

22. Institute for Structural Analysis 22 REC 2008 Deterministic fundamental solution (3) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples εsεs reference axis m θ2θ2 above bond joint: strain distribution bond joint strain of a layer ε(θ 2 ) = ε m – dφ m /dθ 1 · θ 2 + ε s below bond joint: ε(θ 2 ) = ε m – dφ m /dθ 1 · θ 2 flexible bond – enhancement of the differential equation system

23. Institute for Structural Analysis 23 REC 2008 Deterministic fundamental solution (4) different materials within the same layer force-strain-dependency of the layer ΔN r = E T,r,m Δ ε r,m b m (θ 2 ) d θ 2 Σ ∫ m θ 2,r ΔM r = E T,r,m Δ ε r,m b m (θ 2 ) θ 2 d θ 2 Σ ∫ m θ 2,r = A(θ 1, z) (n-1) · Δz(θ 1 ) + Δb(θ 1, z) dΔz(θ1)dΔz(θ1) d(θ 1 ) (n) [k] (n) [k] (n) [k-1] + d(θ 1 ) (n-1) · Δz 1 (θ 1 ) + m(θ 1 ) (n-1) · Δz 1 (θ 1 ) (n) [k] (n) [k] ··· number of materials within the layerm.. number of layers at the cross sectionr.. Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

24. Institute for Structural Analysis 24 REC 2008 θ 1, u θ 2, v 10 m10 p g concrete g steel p = 400 kN/m (~ 60% ultimate load) system and loading Example 1: steel-concrete composite girder (1) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

25. Institute for Structural Analysis 25 REC 2008 Example 1: steel-concrete composite girder (2) 1600 mm 160 180 x 10 1000 x 10 300 x 16 (span area) 400 x 30 (support area) concrete:C 35/45 steel:construction steel S 355 reinforcement steel BSt 500 cross section material parameters E = 33500 N/mm² f cm,cyl = 43 N/mm² E = 210000 N/mm² f y = 360 N/mm² E = 210000 N/mm² f y = 500 N/mm² Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

26. Institute for Structural Analysis 26 REC 2008 slip s [mm] 1.53.00 400 shear stress–slip dependency 800 1200 1600 shear stress σ 12 [kN/m²] 3 cases nonlinear flexible bond (fuzzy process) linear flexible bond (fuzzy process) rigid bond (deterministic) Fuzzy material parameters Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

27. Institute for Structural Analysis 27 REC 2008 displacement v [mm] 10300 0.25 0.5 0.75 1 μ 3 cases nonlinear flexible bond (fuzzy process) linear flexible bond (fuzzy process) rigid bond (deterministic) 4.25 m v 20 Fuzzy result: vertical displacement v Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples g + p

28. Institute for Structural Analysis 28 REC 2008 Example 2: framework 4 m 300 mm reinforcement 10 Ø 12 stirrup Ø 6, a = 250 mm 3 m 300 64 p + g H K K concrete C20/25 400 damage zone K load and modification process Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples 1 2 34

29. Institute for Structural Analysis 29 REC 2008 Example 2: fuzzy parameters increments relievingreloading, failureloading load and modification process damage K [10 3 kNm/rad] Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples 20 40 60 80 100 120 140 160 180 200 4000 3000 2000 1000 0 17100 15000 13700 6860 6000 5490 40000 35000 32000 36 28 23 σ [N/mm 2 ] ε 36 28 23 0 0.0019 0.004 μ 1.0 load factor k k 0.9 1.0 1.1 μ 1.0

30. Institute for Structural Analysis 30 REC 2008 Example 2: fuzzy results relieving reloading, failureloading load and modification process damage Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples 20 40 60 80 100 120 140 160 180 200 increments fuzzy damage indicator 1.0 0.8 0.6 0.4 0.2 0 I D = 1 – [λ( τ )/ λ pl ( τ )]

31. Institute for Structural Analysis 31 REC 2008 Example 2: fuzzy results Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples 1.0 0.75 0.5 0.25 0 20 40 60 80 failure load H [kN] μ

32. Institute for Structural Analysis 32 REC 2008 Motivation Modification of systems Fuzzy functions Fuzzy structural analysis Examples cross sections and system tubular section (centrifugally cast column) OD 250 L = 8000 mm 10 ID 80 mm concrete C30/37 steel 10 Ø 16 fine concrete with textile reinforcement 3 layers with 80 rovings each P crit Example 3: textile-strengthened column

33. Institute for Structural Analysis 33 REC 2008 Material parameters concrete C 30/37 E = 32000 N/mm² f cm,cyl = 38 N/mm² reinforcement steel BSt 500 E = 210000 N/mm² f y = 500 N/mm² fine concrete C 30/37 E = 30000 N/mm² f cm,cyl = 81 N/mm² textile ARG 1100-01 E = 81170 N/mm² f u = 574 N/mm² load and modification process stress problem P increments P crit P serv system modification Motivation Modification of systems Fuzzy functions Fuzzy structural analysis Examples

34. Institute for Structural Analysis 34 REC 2008 Results: critical load P crit (physically nonlinear) 6008001000120014001600 P [kN] transversal displacement at L/2 [mm] strengthened column, modification at P serv = 400 kN strengthened column, modification at P serv = 200 kN strengthened column, modification at unloaded state unstrengthened column (incrementally applied) 5 0 P crit = 1200 kN P crit = 1210 kN P crit = 1220 kN P crit = 930 kN Motivation Modification of systems Fuzzy functions Fuzzy structural analysis Examples

35. Institute for Structural Analysis 35 REC 2008 historical view current view Example 4: Syratalbrücke Plauen (Germany) Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

36. Institute for Structural Analysis 36 REC 2008 90 m 17 m Example 4: natural stone arch bridge Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

37. Institute for Structural Analysis 37 REC 2008 uncertain nonlinear node springs linear support springs non-deformated arch deformated arch horizontal component of the arch deformation Example 4: natural stone arch bridge Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

38. Institute for Structural Analysis 38 REC 2008 masonry strength modification process system modification masonry rehabilitation (grouting) intrinsic time load intrinsic time traffic load relievingreloadingloading dead load tempe- rature traffic load dead load (deck ) load process Modification and load process Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

39. Institute for Structural Analysis 39 REC 2008 σ [N/mm² ] ε εuεu ε rehabilitation original constitutive law modified constitutive law Masonry rehabilitation Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples 22.3 17.8 0 = 0.0038 0.006

40. Institute for Structural Analysis 40 REC 2008 μ fKfK stiffness factor of nonlinear springs 1 0.91.01.3 σ old μ unrehabilitated masonry strength 1 1117.825 [N/mm²] σ new μ rehabilitated masonry strength 1 1822.329 [N/mm²] ftft μ factor of traffic load 1 0.851.01.51.3 Fuzzy input parameters Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

41. Institute for Structural Analysis 41 REC 2008 η – failure load factor (traffic load) ~ 0.75 (η)(η) 1 0.5 0.25 0 failure load factor η 1234567 εuεu ε reh. σ ε σ σ ε ε 123 1 2 3 without system modification unrehabilitated masonry strength case 1: system modification unrehabilitated/rehabilitated masonry strength case 2: without system modification rehabilitated masonry strength case 3: Fuzzy result: failure load factor Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

42. Institute for Structural Analysis 42 REC 2008 Fuzzy result values v – vertical displacement of the crown of the arch ~ 10 0.75  1 0.5 0.25 0 v [cm] ABC load and modification process 40 30 20 points of the load and modification process load and modification intrinsic time system modification masonry rehabilitation (grouting ) A B C Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

43. Institute for Structural Analysis 43 REC 2008 Analyzing a structure close to reality requires to consider the complete load and modification process. The parameters of the load and modification process are generally uncertain. They may be described by fuzzy processes. Conclusion Motivation Modification of structures Fuzzy functions Fuzzy structural analysis Examples

44. Institute for Structural Analysis 44 REC 2008 Dresden University of Technology Dresden Source City of Dresden, 2008 Thank You

45. Institute for Structural Analysis Uncertain processes and numerical monitoring of structures Wolfgang Graf Bernd Möller Matthias Bartzsch www.tu-dresden.de/isd www.uncertainty-in-engineering.net GeorgiaTech Savannah