PSI - Issue 78

Marco Terrenzi et al. / Procedia Structural Integrity 78 (2026) 418–425

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The most widely used model is Rayleigh damping, where the damping matrix, denoted as c , is directly proportional to a linear combination of the mass and sti ff ness matrices: c = α · m + β · k (1) where m is the mass matrix, k is the sti ff ness matrix, and α and β are the Rayleigh damping coe ffi cients determined from the damping ratios specified for two selected modes or two frequencies, i and j . The two modes or frequencies are defined to ensure that damping in all significant response modes remains near the target value., since values outside the range lead to higher damping ratios and thus would be overdamped.Rayleigh damping traditionally uses the initial sti ff ness ( k initial ). However, there have been recent proposals to update this sti ff ness as the structure sustains damage, for instance, by employing the current (tangent) or committed sti ff ness. Charney (2008) highlighted that using the initial sti ff ness can lead to excessive damping forces, suggesting the tangent sti ff ness as a more appropriate alternative. However, Chopra and McKenna (2016) caution against using tangent sti ff ness, arguing that if it becomes negative due to softening, the physical meaning of the damping matrix c could be lost. Modal damping is another modeling approach that uses a linear combination of mass and sti ff ness damping. This method allows you to assign a specific damping ratio to any vibration mode, and it can be expressed as: c = m    N n = 1 2 ξ n ω n M n φ n φ T n    m (2) where ξ n is the damping ratio at the n th mode, φ n is the n th mode eigenvector and M n is the n th mode generalized mass. The majority of Italy’s reinforced concrete building heritage originates from the 1960s and 1970s (ISTAT , 2011), having been constructed either without seismic provisions (Pancottini et al., 2025) or in accordance with obsolete seismic codes. This paper analyzes a six-story case study building, representative of a considerable percentage of existing building (Basaglia et al., 2021).The structural layout (Fig. 1) includes four seven-bay frames in the X-direction and only two external frames in the Y-direction, with a consistent inter-story height of 3.2 m. The alignment of most beams and all column strong axes along the X-direction leads to significantly higher lateral sti ff ness and strength in the X-direction compared to the Y-direction.Consistent with the design practices of the 1970s, the dimensions and rebar configurations of the columns and beams were determined according to the allowable stress design method (Barbagallo et al., 2023; Cantagallo et al., 2023). For material properties, the concrete has a characteristic cubic compressive strength of Rck = 25[ MPa ], and the reinforcing steel is FeB 38 K , exhibiting a yield stress of fyk = 375[ MPa ]. The building is modelled without (Bare Frame or BF) and with infills (IF).The contribution of infills is often neglected in nonlinear models, despite IF buildings exhibiting distinct structural behavior compared to BF buildings. Building models were generated in OpenSees using the STKO pre / post-processor (Petracca et al., 2017). A distributed plasticity approach was used to model both beams and columns (Terrenzi et al., 2020), using the Beam-with-Hinges (BWH) element (Scott and Fenves, 2006). This element restricts nonlinear constitutive behavior to defined plastic hinge regions ( L pl ), which are modeled as fiber sections (Spacone et al., 1996a,b). Conversely, the element’s central portion is represented by elastic sections, carefully calibrated to account for cracking.The fiber sections model con crete using the Concrete01 material model (Kent and Park, 1971; Scott, 1982). For this, the same properties are applied to both confined and unconfined concrete due to the stirrup spacing being too wide for e ff ective confinement. Steel is modeled using the Steel01 material model (Pinto, 1973). The L pl is assumed to be equal to the section depth; for rectangular column sections, the average of the two depths is used. P-Delta e ff ects are included in the models. Floor diaphragms are simulated using the ”rigidDiaphragm”, which inherently restrains axial beam deformation and can induce spurious axial forces. To mitigate this issue, axial bu ff er elements (Barbagallo et al., 2020) were added at beam ends. Given that this paper focuses on damping, the models do not account for potential brittle failure mechanisms 2. Case study description 2.1. Numerical models and modal results