3.2 Illustration We applied the procedure on the nominal (2612 spot welds) and optimized (2238 spot welds) designs in order to quantify the robustness and analyze their relative behavior as a function of increasing number of missing of the most influential spot welds. The robustness curves were obtained due to missing of up to 100 spot welds by removing 10 spot welds of highest indicator values in each iteration. The number of RSWs removed in each iteration may be increased or decreased considering the trade-off between the total number of spot welds to be checked and the time required by one numerical analysis. The robustness curves obtained for both modes are shown in fig 7 for both designs. The eigenfrequencies of the nominal design were taken as references to calculate the relative eigenfrequency shifts. The curves illustrate that optimized design is relatively less sensitive to the missing of spot welds despite of the fact that it contains 14% less number of spot welds. In other words, it can afford to loose more number of most influential spot welds before violating a specified level of degradation in the eigenfrequencies: for example to observe a 6.0% relative eigenfrequency shift for mode 1, optimized design can loose up to 100 most influential spot welds while nominal design requires only 40 spot welds. This gain in robustness is achieved in the optimization process by adding the spot welds on the sensitive regions in the proximity of critical spot welds. In turn, impact of the critical spot welds decreased, reflected by the decreases in indicator values of these critical spot welds as observed in fig 6. 20 40 60 80 100 −8 −6 −4 −2 0 2 α − Number of RSWs Relative Eigenfrequency Shift (%) Nominal design Optimized design (a) 20 40 60 80 100 −2 −1.5 −1 −0.5 0 0.5 α − Number of RSWs Relative Eigenfrequency Shift (%) Nominal design Optimized design (b) Fig. 7 Robustness curves for two modes (a)mode 1 (b)mode 2 However, note that most of the degradations in the eigenfrequencies are due to only the first 30 RSWs for both designs. This implies that these are the most critical RSWs and need special attention of the designer. This is a valuable feedback and designer can use this auxiliary information in various ways such as: Robust optimization and quality control in spot welded structures 305
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