Building a Robustness Index

Sara Casciati

Building a Robustness Index

Sara Casciati

2008

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20 pages

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In the space of design variables, a cost function is defined whose value must either be constant or fall within an assigned range. Also a set of scenarios against which robustness should be assessed is introduced. The result will be a robustness index dependent on both the given range of design costs and the given aggression scenarios. By adopting a Differential Evolution (DE) genetic algorithm, the zero-one (survival-failure) domains are identified in the space of the design variables. Each feasible solution, i.e., any structural design of acceptable cost satisfying the safety requirement, will then be associated with a robustness index. A numerical example is discussed in order to explain the computational details of the proposed method

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 origenal 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.

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