Uncertainty modelling using software FReET D. Novak, M. Vorechovsky, R. Rusina Brno University of Technology Brno, Czech Republic 1/30
Outline Introduction Methods and main features Software FReET Selected applications Probabilistic analyses of concrete structures Statistical size effect studies Identification of computational model parameters 2/18 2/25 2/30
Methods and main features Small-sample simulation of Monte Carlo type Imposing statistical correlation Sensitivity analysis Reliability analysis 2/18 12/25 3/30
Latin hypercube sampling The range (0; 1) of PDF Φ(Y i ) of each random variable Y i is divided into N non-overlapping intervals of equal probability 1/N (McKay et al. 1979. Iman & Conover 1980, Iman & Shortencarier 1984). The centroids are selected randomly based on random permutations of integers. Every interval of each variable is used only once during the simulation process. variable simulation 2/18 13/25 4/30
LHS: Step 1 - simulation x i, k Sim ( ) b = N x f x dx a Huntington & Lyrintzis (1998) Mean value: accurately Stand. deviation: significant improvement 2/18 14/25 5/23
LHS: Step 2 imposing statistical correlation simulation variable x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 x 5 y 5 z 5 x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 8 x NSim y NSim z NSim Simulated annealing: Probability to escape from local minima Cooling - decreasing of system excitation Boltzmann PDF, energetic analogy r kb ( ) P E e E T 2/18 15/25 6/30
LHS: Step 2 imposing statistical correlation variable x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4 simulation x 5 y 5 z 5 x 6 y 6 z 6 x 7 y 7 z 7 x 8 y 8 z 8 x NSim y NSim z NSim 2/18 16/25 7/30
Sensitivity analysis Nonparametric rank-order correlation between input variables ane output response variable Kendall tau Spearman ( q,p ), j = 1,2, K N τi = τ ji j, r s = ( n 1)( n + 1) Robust - uses only orders Additional result of LHS simulation, no extra effort Bigger correlation coefficient = high sensitivity Relative measure of sensitivity (-1, 1) 6 i= 1 1 n n d 2 i INPUT x 1,1 x 1,N INPUT q 1,1 q 1,N OUTPUT R 1 R, N OUTPUT p 1 p N 2/18 17/25 8/30
Reliability analysis Simplified rough estimates, as constrained by extremally small number of simulations (10-100)! Cornell safety index Curve fitting FORM, importance sampling response surface 2/18 18/25 9/30
FREET Probabilistic techniques Crude Monte Carlo simulation Latin Hypercube Sampling (3 types) First Order Reliability Method (FORM) Curve fitting Simulated Annealing Bayesian updating Response/Limit state function Closed form (direct) using implemented Equation Editor (simple problems) Numerical (indirect) using user-defined DLL function prepared practically in..any programming language (C++, Fortran, Delphi, etc.) General interface to third-parties software using user-defined *.BAT or *.EXE http://www.freet.cz 2/18 23/25 10/30
FREET Stochastic model (inputs) Friendly Graphical User Environment (GUE) 30 probability distribution functions (PDF), mostly 2-parametric, some 3-parametric, two 4-parametric (Beta PDF and normal PDF with Weibullian left tail) Unified description of random variables optionally by statistical moments or parameters or a combination of moments and parameters PDF calculator Statistical correlation (also weighting option) Categories and comparative values for PDFs Basic random variables visualization, including statistical correlation in both Cartesian and parallel coordinates http://www.freet.cz 2/18 23/25 11/30
FREET Stochastic model 2/18 23/25 12/30
FREET Correlation, simulation 2/18 23/25 13/30
FREET Assessment 2/18 23/25 14/30
Software tools: SARA Studio Probabilistic software FReET http://www.freet.cz + Software for nonlinear fracture mechanics analysis ATENA 2/18 21/25 15/30
ATENA Well-balanced approach for practical applications of advanced FEM in civil engineering Numerical core state-of-art background User friendly Graphical user environment visualization + interaction 2/18 22/25 16/30
Material models for concrete: ATENA software Numerical core advanced nonlinear material models concrete in tension tensile cracks post-peak behavior smeared crack approach crack band method fracture energy fixed or rotated cracks crack localization size-effect is captured 2/18 20/25 17/30
Software tools: SARA Studio 2/18 24/25 18/30
Probabilistis analyses of concrete structures Coll de Isarco viaduct 30 Safety index β 25 20 15 10 CoV = 0.15 CoV = 0.12 CoV = 0.10 CoV = 0.075 CoV = 0.05 Eurocode Design Load 5 0 0 25 50 75 100 125 150 175 200 225 250 Line Load [kn/m] 2/18 9/25 19/30
Probabilistis analyses of concrete structures Coll de Isarco viaduct Random variable description Symbol Units Mean value COV Distribution type Reference Concrete grade B500 6 Modulus of elasticity Ec GPa 36.95 0.15 Lognormal Poisson's ratio µ - 0.2 0.05 Lognormal Estimation 6 Tensile strength ft MPa 3.257 0.18 Weibull 6,7 Compressive strength fc MPa 42.5 0.10 Lognormal 8 Specific fracture energy Gf N/m 81.43 0.20 Weibull 6 Uniaxial compressive strain ε c - 0.0023 0.15 Lognormal Reduction of strength cred - 0.8 0.06 Rectangular Estimation Critical comp displacement wd m 0.0005 0.10 Lognormal Estimation Specific material weight ρ MN/m 3 9 0.023 0.10 Normal Prestressing strands 10 Modulus of elasticity Es GPa 200.0 0.03 Lognormal 10 Yield stress fy MPa 1600.0 0.07 Lognormal 9 Prestressing force F MN 21.85 0.04 Normal Area of strands As m 2 9 0.0237 0.001 Normal 2/18 9/25 20/30
Probabilistis analyses of concrete structures soil structure interaction Stability of concrete tunnel tube in complicated geological conditions Influence of spatial variability of Young modulus and material constants of Drucker-Prager criterion (based on cohesion and angle of internal friction) Analyzed part 50 x 60m, diametr of tunnel 11m, wall thickness 0.5m Plain strain state, 5000 finite elements 5.657E-02 6.300E-02 7.350E-02 8.400E-02 9.450E-02 1.050E-01 1.155E-01 1.260E-01 1.365E-01 1.470E-01 1.575E-01 1.680E-01 1.718E-01 2/18 9/25 21/30
Statistical size effect studies Malpasset dam (failed 1959) Calculation for different sizes, microplane M4 model Bažant, Vořechovský, Novák - Icossar 2005 2/18 9/25 22/30
Statistical size effect studies 2/18 9/25 23/30
Statistical size effect studies Shear failure, verification of design formulas 300 1000 925 75 500 50 450 300 250 225 25 300 110 125 15 300 BN100 BN50 BN25 BN12 Nominal strength [kn/m2] 1600 1400 1200 1000 800 Experiments Statistical simulation ACI Eurocode 2 Collins Bažant 600 400 0.0 0.2 0.4 0.6 0.8 1.0 Size (depth of beam) [m] 2/18 9/25 24/30
Identification of computational model parameters Structural response Material model parameters Stochastic calculation (LHS) training set for calibration of synaptic weights and biases 2/18 9/25 25/30
Identification of computational model parameters shear wall test Variable Symb ol Unit Mean value COV Modulus of elasticity E GPa 30 0.10 Tensile strength ft MPa 2.5 0.10 Compressive strength fc MPa 30 0.10 Fracture energy GF N/m 75 0.20 Compressive strain εc - 0.0025 0.20 Max. comp. displacement Bilinear diagram of steel for smeared reinforcement wd m 0.003 0.30 x1 m 0.0027 0.10 fx1 kn 574 0.10 x2 m 0.015 0.10 fx2 kn 764 0.10 2/18 9/25 26/30
Identification of computational model parameters 2/18 9/25 27/30
Identification of computational model parameters Horizontal force [kn]. 1000 800 600 400 200 0 S2 Experiment ATENA 6 parameters identified ATENA 10 parameters identified 0 2 4 6 8 10 12 14 Horizontal displacement [mm] DLNNET 6 par. 10 par. E [MPa] 29,9 33,0 f t [MPa] 2,47 2,47 f c [MPa] 34,51 35,3 G f [MN/m] 75,0 77,85 e c [-] 2,51E-03 2,57E-03 w d [m] 3,00E-03 3,10E-03 x 1 2,72E-03 2,74E-03 fx 1 566,9 570,7 x 2 1,50E-02 1,47E-02 fx 2 764 768,8 2/18 9/25 28/30
Conclusions FREET - software for statistical, sensitivity and reliability analysis Suitable for analysis of computationally intensive problems (eg. continuum mechanics, FEM) Software development: 1) Stand alone module - definition of reliability problem (user-defined response/limit state function) 2) Integration with software ATENA - nonlinear fracture mechanics of reinforced concrete structures (Červenka Consulting) 2/18 25/25 29/30
Thank You for Your attention! http://www.freet.cz 2/30