40 B. T. Le et al. Fig. 8 Objective functionJ(∆θi)∗ through iterations Table 2 Parameter values before and after finite element updating Paremeter Initial value Updated value Change ρsteel 7850 kg/m3 7039 kg/m3 -811kg/m3 k1 10000 kN/m 20736kN/m 10736kN/m k2 10000kN/m 20736kN/m 10736kN/m Esteel 210GPa 218.9GPa 8.9 GPa Table 3 Natural frequencies and MAC values before and after updating Mode Frequency, f[Hz] MACvalue Measured Initial Error[%] Weighted errors[%] Updated Error[%] Weighted errors[%] Initial Updated Change 1 3.15 2.48 -21.27 -11.92 3.18 0.95 0.53 0.957 0.981 0.024 2 4.42 3.42 -22.62 -6.44 4.26 -3.62 -1.03 0.953 0.977 0.024 3 8.83 7.95 -9.97 -0.71 8.63 -2.27 -0.16 0.649 0.664 0.015 4 10.19 10.1 -0.88 -0.05 10.84 6.38 0.34 0.960 0.958 -0.002 5 13.52 14.39 6.43 0.20 15.2 12.43 0.38 0.699 0.703 0.004 Absolute average 10.2 4.3 0.70 0.71 The updating process yielded a commendable reduction in average frequency errors from 10.2%to 4.3%. Notably, the most significant improvements were observed for the first two modes, aligning with the amplified weighting assigned to these modes. A slight increase in the MAC value suggests a closer approximation of the FE model’s mode shapes to those extracted from measurements. Sensitivity Analysis of Lifting Towers One of the motivations for performing studies at Magerholm as a case study stems from the collapse of its ferry dock during Ingunn storm, one of the most powerful storms to impact Norway in 30 years [17], on 31st January 2024. Following the storm, the entire deck of the structure detached from the lifting towers and sank into the water, necessitating a salvage operation and reattachment to the towers. Although the exact cause of the collapse remains uncertain, the authors suspect that similar damages can be caused by failure of the lifting towers. Consequently, the authors conducted a sensitivity analysis on the updated model of lifting tower stiffness to assess its impact on the dynamic behavior of the linkspan. The results are presented in Figs. 9 and 10. Figure 9 captures the sensitivity of frequencies when changing the stiffness of one tower and keep the other one constant at k =20736kN/mwhich is the stiffness of the towers after updating. On the other hand Fig. 10 capture that when changing the stiffness of both towers at the same time. As expected, there is a clear correlation between tower stiffness and modal behavior. Most modes are sensitive to stiffness variations, except for mode 6, which remains relatively unaffected. The first bending mode (mode 1) is particularly sensitive to stiffness changes, while modes 3 and 6 show minimal sensitivity. Interestingly, the first torsional mode (mode 2) exhibits
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