INVERSE RELIABILITY ANALYSIS IN STRUCTURAL DESIGN

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1 INVERSE RELIABILITY ANALYSIS IN STRUCTURAL DESIGN David Lehký, Drahomír Novák Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic.6.0 ISUME 0, Prague, Czech Republic

2 Introduction Inverse analysis in structural design determination of values of design parameters (proportions, reinforcement, material properties, etc.) to satisfy particular limit state (both ultimate and serviceability). Uncertainties and randomness: partial safety factor design (semi-probabilistic design) E d (X i,k, γ i, ) R d (X j,k, γ j, ) Inverse analysis analytical trial and error method Design parameters are deterministic. fully probabilistic design.6.0 ISUME 0, Prague, Czech Republic

3 fully probabilistic design Safety margin: Z = R E generally: Z = g(x) g(.) limit state function Introduction and motivation Reliability measures: Failure probability: p f = P(Z < 0) Reliability index: β = -Φ - (p f ) Inverse analysis advanced methods, e.g. ANN + stochastic analysis Design parameters are deterministic or random variables (statistical moments)..6.0 ISUME 0, Prague, Czech Republic 3

4 Inverse reliability problem formulation basic random variables: X = X, X,, X j,, X n deterministic design parameters: d = d, d,, d k,, d p design parameters of random variables: r = r, r,, r l,, r q safety margins Z j with target failure probabilities p f,j Inverse problem: Given: p f,j Find: d or/and r Subject to: Z j = g(x, d, r) j = 0 for j =,,, m. Design parameters alternatives: Variable Deterministic Random Mean Std. d k? r l? prescribed r l prescribed? r l??.6.0 ISUME 0, Prague, Czech Republic 4

5 Feed-forward multilayer network (backpropagation type) NEURON: NEURAL NETWORK: Output from neuron: y ( x) = f ( w p ) + b = f k k k (input of ANN, hidden layers, output layer) k number of input impulse (,...,K) w k weight coefficient of connecting path from k-th neuron of previous layer p k impulse from k-th neuron previous layer b bias of neuron f transfer function of neuron.6.0 ISUME 0, Prague, Czech Republic 5

6 Artificial neural network Behavior of ANN is determined by: number of hidden layers and neurons in them synaptic weights conductivity of connecting paths biases transfer (activation) functions (binary, linear, nonlinear neurons) Types of transfer functions: a) two-valued function b) linear transfer function c) hyperbolic tangent (symmetric sigmoid function) d) sigmoid function.6.0 ISUME 0, Prague, Czech Republic 6

7 Training of artificial neural network active period (simulation of process) ANN activities adaptive period (training) Training of neural network: training set, i.e. ordered pair [p i, y i ] input and output vector Minimization of criterion: E = N K ( v * y ) ik yik i= k = N number of ordered pairs input - output in training set; y ik * required output value of k-th output neuron at i-th input; y ikv real output value (at same input)..6.0 ISUME 0, Prague, Czech Republic 7

8 ANN based inverse reliability analysis.6.0 ISUME 0, Prague, Czech Republic 8

9 Software tools FReET: Simulation and reliability IREL: Inverse reliability DLNNET: Neural networks.6.0 ISUME 0, Prague, Czech Republic 9

10 Example A limit state function with single design parameter θ. Target reliability index β =.0. g = exp[ θ ( u 3 )] + u + u3 u Random variables: Variable Distribution Mean Std COV u Normal 0 -- u Normal 0 -- u 3 Normal 0 -- u 4 Normal 0 -- θ Lognormal ( par)? Design parameter and its randomization (LHS): Variable Distribution Mean Std a b mean(θ) Rectangular ISUME 0, Prague, Czech Republic 0

11 Artificial neural network: Example Training set : 4 LHS simulations of mean(θ) multiple FORM analyses with each LHS simulation β Results: mean(θ) β β target ISUME 0, Prague, Czech Republic

12 Example A set of three limit state functions g, g, g 3 with target reliability indexes β = 3.0, β = 3.5, β 3 = 4.0. g g g 3 = x = x = x 4x x x 4 x 4 x x x x x 4 Random variables: Variable Distribution Mean Std COV x Normal 6? -- x Lognormal ( par)? x 3 Lognormal ( par)? x 4 Gumbel max EV Design parameter and its randomization (LHS): Variable Distribution Mean Std a b std(x ) Rectangular mean(x ) Rectangular mean(x 3 ) Rectangular ISUME 0, Prague, Czech Republic

13 Example Artificial neural network: Training set : 00 LHS simulations multiple FORM analyses with each LHS simulation β Results: std(x 3 ) mean(x ) mean(x 3 ) β (β,target ) β (β,target ) β 3 (β 3,target ) (3.0) (3.5) (4.0).6.0 ISUME 0, Prague, Czech Republic 3

14 Example 3 The aim is to design proportions of rectangular cross-section (mean values of width b and height h) of timber beam to satisfy reliability level given in Eurocodes. Two limit states are considered: () ultimate limit state reliability index β = 3.8. () serviceability limit state reliability index β =.5. g g = = M u R lim, fin M E u net, fin M M R E = θ = θ R E 6 8 bh ( g + q) l.6.0 ISUME 0, Prague, Czech Republic 4 k mod f m u u u u lim, fin net, fin, fin, fin = = l = 00 = θ E ( u + u ), fin 4 gl E bh 4 ql E bh 3, fin 3 ( + k ), def ( + k ), def

15 Random variables: Example 3 Variable Distribution Mean Std COV l [m] Normal b [m] Normal? h [m] Normal? E [GPa] Lognormal ( par) f m [MPa] Lognormal ( par) g [kn/m] Gumbel max EV q [kn/m] Gumbel max EV θ R [-] Lognormal ( par) θ E [-] Lognormal ( par) Design parameter and its randomization (LHS): Variable Distribution Mean Std a b mean(b) Rectangular mean(h) Rectangular ISUME 0, Prague, Czech Republic 5

16 Example 3 Artificial neural network: Training set : 00 LHS simulations multiple FORM analyses with each LHS simulation β Results: mean(b) mean(h) β (β,target ) (3.8) β (β,target ).500 (.5) Design: b = 40 mm h = 0 mm β = β = ISUME 0, Prague, Czech Republic 6

17 Summary + Methodology for inverse reliability analysis: Artificial neural network + stochastic analysis (LHS simulation method) + Helpful in case of fully probabilistic design + Design parameters deterministic as well as random ones + Multiple design parameters problem as well as multiple limit state function problem can be solved + Statistical correlation among variables can be imposed + Sotware tools ready for routine applications and we welcome interesting problems for collaboration Thank you for paying attention!.6.0 ISUME 0, Prague, Czech Republic 7