PSI - Issue 44

P. Morandi et al. / Procedia Structural Integrity 44 (2023) 1060–1067 Author name / Structural Integrity Procedia 00 (2022) 000–000

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3.2. Retrofit system modelling The modular steel frames were numerically reproduced by explicitly modelling each component as a one dimensional beam finite element (FE) with an isotropic elastic behaviour, with no failure limit. This simplified modelling assumption was deemed reasonable since experimental evidences demonstrated that the retrofit behaviour was controlled by the failure of the retrofit-to-masonry anchors rather than the attainment of the axial or flexural strength of steel components. Cross-sectional area and moments of inertia were properly assigned reproducing the actual stiffness of the steel members. Rigid links were defined to connect adjacent modular frames. To correctly capture the retrofit influence on the pier lateral response, an accurate modelling strategy was required to characterize the connections between the FE frames and the masonry block assembly. Specifically, as shown in Fig. 4, structure deformable links were defined in correspondence of the actual position of anchors. A shear-yield model was assigned to each link for the translational degrees of freedom in directions x and z , while a normal-yield model was assigned for the translational degree of freedom (DOF) in direction y . For each direction, each structural links was defined by a stiffness k and by a yielding strength F yield , which were calibrated against the experimental results of the characterization tests performed on retrofit-to-masonry anchors. Specifically, a value of k of 5∙10 6 N/m was assigned to x and z translational DOF while a value of 2.5∙10 7 was assigned in the y direction. A value of F yield = 11 kN was equally assigned to links in x , y and z directions.

Fig. 4. Modelling of the retrofit frame-to-masonry anchors by means of three-dimensional structural links.

3.3. Numerical results The computational procedure of 3DEC is based on a dynamic time-integration algorithm that solves the equations of motion by an explicit finite difference method. Static problems can be solved within the same algorithm adopting an approach conceptually similar to dynamic relaxation (Otter et al. 1966). Specifically, through a numerical servo mechanism, named adaptive global damping (Cundall 1982), the equations of motion are damped to quickly reach a force equilibrium state. Size, density and time-scaling techniques were also employed according to Malomo et al. (2019) to obtain an acceptable compromise between the accuracy of results and the computational effort. The fully fixed RC foundation and the RC top beam were modelled as an assembly of linear elastic FD regions ( E = 40000 MPa). Equivalent densities were also assigned to the top beam to reproduce the actual imposed compressive load. The comparisons between experimental and numerical hysteretic curves, reported in Fig. 5, clearly demonstrate how numerical models are able to simulate with very good approximation the experimental cyclic response of the specimens, with the exception of the maximum force, which was found a bit smaller in the analyses due to the current inability to capture the tension component of the lateral response of this masonry typology. It is worth to notice that numerical models correctly predict not only the elastic stiffness and, in a lesser extent, the maximum lateral strength, but also the pier energy dissipation capacity as well as the progressive stiffness/strength degradation as drift ratio increases. This aspect is of particular importance considering that the investigated retrofit solution allowed the pier undergoing to higher drift ratios without increasing its lateral strength. Moreover, Fig. 6 shows for both specimens the comparison between experimental and numerical cracking patterns at the end of the tests. The DE models satisfactorily reproduced also the experimental damage mechanisms. For numerical models, damage