PSI - Issue 78

Stefano Bracchi et al. / Procedia Structural Integrity 78 (2026) 745–752

746

1. Introduction The seismic demand in historical masonry buildings is usually evaluated by neglecting the effects owing to the foundation compliance, i.e. the elongation of the structural period and the energy dissipation associated to the oscillation motion of the foundation. To account for soil-foundation-structure interaction (SFSI), in a decreasing scale of complexity, the first modelling option is to create a detailed numerical model of the structure, the foundation and the geological media. Such approach is however computationally expensive and not adequate for a complete seismic risk analysis. An alternative is represented by the Macro-Element (ME) theory, in which the soil foundation is replaced by a unique lumped plasticity model, capable to describe the incremental relationships between generalized forces and work-conjugated displacement, by appropriate nonlinear constitutive equations. The advantage of the ME theory over the complete analysis of the soil-foundation-structure system is manifold (Nova and Montrasio 1991, Gottardi et al. 1999, Houlsby and Cassidy 2002, Salciarini and Tamagnini 2009, Marchi et al. 2011, Pecker et al. 2014, Pisanò et al. 2016). Moreover, ME can be coupled with multi-axial viscous dampers, to account for energy dissipation by radiation, and fictitious masses, to account for frequency dependence of the foundation behavior. At low strains, ME behave like multi-axial springs so that the above assembly reduces to a set of uncoupled Kelvin-Voigt models and fictitious masses. Such reduced model is far to be new and is referred to in the literature as LPM or Lumped Parameter Model (Wolf 1991). LPMs can be thought of as a reliable choice to study the response of buildings to low and moderate intensity level earthquakes. The focus is set in this work on nonlinear dynamic analyses of unreinforced masonry (URM) buildings including LPMs. Among the various modelling strategies, the equivalent-frame approach is adopted, due to its low computational burden and accuracy of results. To account for SFSI, the approach proposed herein is to attach a lumped model at the base of each masonry panel, by neglecting the interaction effects among foundations of different panels. For each masonry panel, foundation impedances are evaluated for vibrational degrees of freedom pertinent to the strong direction of the panel, that is along the longitudinal and vertical direction and about the transversal direction. By contrast, the reactions of panels in the out-of-plane direction are not considered. The flexural stiffness of the beam foundation supporting the panels is neglected, while the element is considered as rigid in the longitudinal direction, that is to impose that the nodes at the base have the same horizontal displacement. This modelling strategy is applied to the case study of a building in the city of Naples (Italy). The discussion is developed based on the results obtained at three different Intensity Measure Levels (IML 2, 4, 6) of earthquakes selected from a complete seismic risk analysis, carried out according to the multi-stripes analysis (MSA) approach (Iervolino et al.. 2018). Particularly, LPMs have been calibrated at each IML by taking the average soil properties mobilized by the passage of seismic waves. The seismic demand calculated by means of nonlinear dynamic analyses is then compared to the seismic capacity, obtained through nonlinear static analyses, allowing to assess the modification of building’s performance due to the consideration of SFSI. It must be emphasized that coupling a nonlinear structure with a visco–elastic foundation is not a fair modelling choice, since the soil–foundation model would be unable to dissipate energy by hysteresis as the superstructure does. Moreover, nonlinearity of structural components yields inevitably to a reduction of natural frequency, and the energy dissipated by the viscous damper at such frequency cannot be but very low. In light of this, the scope of this work is just to see how the inertial behavior of the soil-foundation system and the ability of the foundation to dissipate energy by radiation can affect the fragility curves of such kind of buildings.

Nomenclature G / G 0

normalized shear modulus of soil

damping ratio of soil shear strain of soil

D

γ

 mobilized average shear strain of soil S a (T 1 ) spectral acceleration at the fundamental period of the building T r return period of the seismic action