PSI - Issue 78

Gennaro Vesce et al. / Procedia Structural Integrity 78 (2026) 936–943

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In the light of the achieved results, it is evident the needs of adopting the dynamic modal analysis as a basic approach to derive the design parameters guiding the evaluation of the seismic demand (i.e. later force profile) of existing masonry buildings with vertical additions in both cases of linear and nonlinear approaches, as also suggested by the standards IBC 2018 and Eurocode 8. 3.2. Inertial force profile Understanding the seismic action profile along the building’s height represent s a key issue for the quantification of the seismic demand. On this topic, this Section evaluates the effects related to different mass ratios and to the modal contributions into quantifying the dynamic parameters guiding the calculation of the inertial forces. As first step, they have been evaluated the seismic actions at i -th story associated with the j -th modal contributions as follows: F ij = m i  ij  j S a (T j ) (1) where m represents the story mass,  the displacement component of the modal vector,  j the modal participating factor and S a (T j ) the spectral acceleration corresponding to the j -th vibration mode given by the response spectrum. Fig. 3 plots the force profiles associated with the three vibration modes as a function of the considered mass ratios. From this figure emerges that, independently from the mass ratio, the first vibration mode has a predominant contribution into seismic actions computation; the second vibration mode becomes more significant as the mass ratio increases while the contribution of the third mode is negligible. To quantify the effects of the different vibration modes, the modal contributions have been combined through a SRSS combination (also suggested by the codes) and compared with those associated with the first vibration mode only (Fig. 4). The SRSS combination accounts for all the vibration modes characterized with a participating mass greater than 5% (i.e., first and second mode in this application). Fig. 4 shows that the contribution of the second vibration mode does not have a significant effect on the lateral seismic action profile, even if their participating masses are relevant. This is due to the circumstance that the second mode is characterized by reduced vibration periods (belonging to the ascending branch of the spectrum), to which are associated low spectral accelerations with respect to those of the plateau. Moreover, the sum of the effects under square root envisaged by the SRSS combination minimizes furtherly the effects of the second mode. Nevertheless, the Fig. 4 shows that the contribution of the second mode increases with increasing mass ratios. To quantify the seismic demand on the structural system - and in particular on masonry lower structure - due to the introduction vertical additions, the base shear (V b ) distributions along the building ’s height are plotted in Fig. 5. They were compared the base shear calculated considering the SRSS combination, the contribution of the first mode only and that relative to the single lower masonry structure. By looking at Fig. 5, it emerges as the contribution of the second vibration mode is substantially negligible into the computation of the base shear: to this aim, Table 1 reports the base shear ratios calculated as ratios between those associated to the first vibration mode and those obtained with the SRSS combination and, as it can be noted, they tends to unit. Instead, the increment of the base shear of the building with CLT vertical addition range between 10-20% with respect to the masonry structure without additional story, depending on the mass ratio. Nevertheless, such an increment may be not significant in terms of increased seismic demand on the structure. In fact, the presence of the vertical additions produces an extra axial-force over the masonry piers (both at first and second story of the building) which is beneficial because increasing their flexural and shear capacity, providing to increase the missed tensile strength of masonry material. In fact, well-designed masonry structures typically work with a compression demand about 1/10-1/20 of its maximum compression strength, meaning that it can withstand increments of axial force as those provided by light vertical additions (Calderoni et al. 2016).