PSI - Issue 62

Gianluca Bottin et al. / Procedia Structural Integrity 62 (2024) 177–184 G. Bottin et al./ Structural Integrity Procedia 00 (2019) 000–000

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between the supporting elements and the beam for the expected rotations. In Fig. 7a it is possible to observe the maximum rotation reached on the left node, equal to about 0.14 radians. In the left roller, the horizontal displacement (due to elongation of beam generated by the big shift of the neutral axis after yielding) is about 65 mm, while in the bottom part of the beam the displacement field depicted in Fig. 7b is represented, where a grid with the size of 50 mm is superimposed to simplify the understanding of the displacement field. It is possible to observe that point A deforms in position A’ with a horizontal displacement of about 175 mm and a vertical displacement of about 35 mm.

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a

Fig. 7. Deformed shape and rotation (represented by arrows) of the beam (a) 3D view of the whole beam; (b) Lateral view of the left portion with reference undeformed configuration and a grid with cell size of 50 mm superimposed and indication of displacement of point A.

To assess the influence of the statistic variability of parameters (Table 1) on the structural response, a probabilistic analysis is performed with a Monte Carlo approach. In this stage of the research a total of 1000 analyses has been performed extracting, for the selected parameters, a set of values according to their probabilistic distributions. The target displacement of all analyses was set to 1500 mm and an adaptive step algorithm was employed. In order to reduce computational time, the analysis was performed step by step monitoring the load multiplier: when it reduces more than 15% respect the peak force, the analysis is interrupted. In the future of the research, it will be possible to perform more analyses, when the parallel version openseesmp including the reliability module will be available. Anyhow, with 1000 analyses the selected values of parameters fairly match the assumed distribution, supporting the validity of the obtained outcomes. The results obtained in terms of load vs. midpoint displacement curves as well as of mean, standard deviation and COV values for the most significant output parameters are reported in Fig. 8. Due to the large COV value of some parameters (especially the hardening ratio), the response is extremely scattered, in particular with reference to the ultimate displacement (ranging from about 200 mm to more than 1500 mm).

Output parameter Elastic stiffness Symbol Unit kN/mm 30.00 3.47 0.116 mm 23.19 3.67 0.158 kN 686.26 73.85 0.108 kN 820.97 95.11 0.116 Displ. at max force mm 625.54 204.17 0.326 Ultimate displ. mm 670.35 209.88 0.313 Mean Std. dev. COV Yield displ.t Yield force Max force

Fig. 8. Output parameter and force-displ. curve obtained by MC simulation: blue curve is the one corresponding to mean values of parameters.