Building a Robustness Index

1st Workshop and 3rd Management Committee (MC) Meeting COST Action TU0601 – Robustness of Structures February 4 and 5, 2008, ETH Zurich, Switzerland Building a Robustness Index Sara Casciati 1) and Lucia Faravelli 2) 1) University of Catania, Italy 2) University of Pavia, Italy Design Philosophy „ „ „ „ Optimal design: minimum cost & adequate performance. Reliability-oriented optimal design: the structural performance is usually judged based on reliability, which must be kept above a certain threshold. One-level optimization: instead of considering the reliability estimate as a self-standing optimization problem, it is included in the cost-benefit analysis by using the Kuhn and Tucker conditions (FKT). Idea: also robustness can be introduced in the optimization problem as a further requirement for an adequate structural performance. System Objectives „ Cost minimization Æ (FC , wC) „ Reliability requirement Æ (-FR , wR ) „ Robustness Æ (-FI , wI ) where the weights wX represent the Lagrange multipliers and increase with the safety margin within which the corresponding requirement is fulfilled. Reliability & Robustness Oriented Optimal Design Minimize: Reliability Estimate (Kuhn and Tucker Conditions) F( x, u1,…, um ) = wC FC ( x) + wR FR ( x) + wKT FKT ( x, u1,…, um ) + wI FI ( x) where: Cost - (Reliability Requirement) - (Robustness Requirement) x is the design parameters vector; ui is the N×1 vector of the transformed random variables from the original space to the standard normal space, i =1,..,m; m is the number of failure modes. Objective 1: Cost from: Rackwitz, R. (2002). Optimization and Risk Acceptability Based on the Life Quality Index, Structural Safety, 24, 297-331. „ For a public structure whose reconstruction is systematic upon failure, which may only occur at the completion of the structure m ⎞ ⎛m ⎞ ⎛ FC ( x ) = C ( x ) + [C ( x ) + K ] ⎜ ∑ Pf i ( x ) ⎟ ⎜1 − ∑ Pf i ( x ) ⎟ ⎠ ⎝ i =1 ⎠ ⎝ i =1 where: „ „ „ C (x) is the design and construction cost of the structure; K = KM + KH is the sum of the direct cost of the structural failure (including both the damage and the debris removal), KM, and the cost of saving human lives, KH ; Pfi is the probability of the i-th failure mode. Objective 2a: Reliability Requirement „ Imposing: Pfi ≤ 10-6 for i=1,…,m (number of failure modes) „ Is equivalent to minimize: m 10−6 − Pfi ( x ) i =1 Pfi ( x ) FR ( x ) = −∑ Objective 2b: Reliability Estimate „ Kuhn and Tucker Conditions: N −1 ⎡ F KT ( x,u1 ,… ,u m ) = ∑ ⎢| g i (ui , x ) | + ∑ u ij ∇g i ( x ) + ∇g ij ui ( x ) i =1 ⎢⎣ j =1 m „ „ ( ) ⎤ ⎥ ⎥⎦ where: gi (ui ,x) is the i-th limit state function, with i=1,..,m. When, for any i, both the contributes to FKT are null, ||ui || represents the value of the reliability index βi for the i-th failure mode, from which one determines Pfi as Φ(- βi) = Φ(-μgi / σgi), being Φ(.) the standard normal cumulative distribution function. Objective 3: Robustness „ Robust structures are those that develop the less catastrophic failure modes first Æ failure modes hierarchy „ Example: “weak beam/strong column” requirement in the design of a building Portal-Frame Numerical Example 2 3 4 h=5m 1 5 h=5m h=5m N = 9 Random Variables Robustness Index Formulation „ Robustness Function: Z = M pc − M pb ≥ 0 „ Corresponding Robustness Index: μ c − μb μZ I= = ≥0 σZ σ c2 + σ b2 „ (μb , σb), (μc , σc) = means and standard deviations of the plastic moments of the beams and of the columns, respectively. Optimization Problem Particularized for the Portal-Frame Example „ x = [μc μb]T is the vector which collects the design parameters, so that : FI ( x ) = − I = „ „ x2 − x1 2 ⋅13.5 C (x) = 3μc+2μb is the design and construction cost; K = KM + KH , where: KM = 105 is the direct cost of the structural failure,(including both the damage and the debris removal); KH = 2.4 106 is the cost of saving human lives, „ Finally, m = 3 is the number of failure modes and Pfi is the probability of the i-th failure mode, computed by considering the following nonlinear limit state functions. Nonlinear Limit State Functions 2 3 4 h=5m 1 5 h=5m h=5m ‰ Sway-mode: g 1= M 1 + min(M 2c , M 2b ) + min(M 4c , M 4b ) + M 5 − hH ‰ Beam-mode: g 2 = min(M 2c , M 2b ) + 2 M 3 + min(M 4c , M 4b ) − hV ‰ Complete mechanism: g 3 = M 1 + 2M 3 + 2 min(M 4c , M 4b ) + M 5 − hH − hV The corresponding gradients must be computed piecewise. Solution via Differential Evolution (DE) Algorithm Advantages: „ Search driven by objective function itself, instead of its gradient „ Independency of the results accuracy from the initial guess „ Few input parameters „ Moderate computational effort with respect to traditional Genetic Algorithms „ Easily adaptable to the solution of different problems DE Solution Algorithm Randomly generated initial population of NP vectors of the design parameters: x є S = search domain NP = 160 Search range: (500, 1) kNm xi (i = 1,…, NP) = a possible candidate to form the next generation Mutant vector: v i = x r 1 + Γ ⋅ ( x r 2 − x r 3 ) Γ = 0.8 Cross-over: w i j = v i j CR = 0.8 w ij = x ij Selection: ε = 10-4 I = 5000 if r a n d e ls e j ≤ C R , or if F(wi)<F(xi) Æ wi else Æ xi Distance current optimal point from previous one >ε Y Nr. Iterations < I N End DE – Input Parameters Initial Population Size, NP Search Range - Upper Bound - Lower Bound Mutation Amplitude, Γ Crossover Constant, CR Tolerance for Convergence, ε Max Iterations Nr., I 160 500 kN m 1 kN m 0.5 0.8 10-4 5000 Objective Function Weights Calibration F( x, u1,…, um ) = wC FC ( x) + wR FR ( x) + wKT FKT ( x, u1,…, um ) + wI FI ( x) where the weights wX represent the Lagrange multipliers and increase with the safety margin within which the corresponding requirement is fulfilled: Results: objectives evaluated at the optimal point μb I Cost Iterations Without 166.50 robustness 154.57 0.63 808.66 4212 With 168.64 robustness 153.80 0.78 813.52 3098 Case μc Reliability Results Failure gi × 10-5 mode Without robustness column 9.91 βi Pfi × 10-5 5.2190 0.0090 beam complete 5.3943 4.5932 0.0034 0.2183 With robustness column 15.4 5.1002 0.0170 beam complete 5.0656 4.6544 0.0204 0.1624 23.84 -10.73 -0.5 -55.43 Conclusions „ The robustness of a structure can be increased by including it in the design optimization problem. Of course, this happens with an increase of the costs. „ The desired hierarchy of failure modes can be achieved by properly adjusting the weights (Lagrangian multipliers) of the different terms in the objective function. Future Developments „ „ „ Define extreme events scenarios and compute the conditional probabilities for each failure mode. Consider random failure in time by introducing out-crossing rates and interest rates. Generalize the process to different types of structural systems.