PSI - Issue 78

Christian Salvatori et al. / Procedia Structural Integrity 78 (2026) 1529–1536

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1. Introduction Unreinforced masonry (URM) buildings represent a significant portion of the global building stock, particularly in historic centers and rural areas. These structures often exhibit substantial seismic vulnerability due to the inherent material weaknesses and the lack of adequate provisions to resist lateral loads, frequently resulting in out-of-plane failure mechanisms. Even when such vulnerabilities are mitigated and a global three-dimensional behavior is promoted, the seismic performance under horizontal actions may remain insufficient. As a result, extensive efforts have been devoted to developing efficient and effective strategies for retrofitting existing URM buildings and designing new ones. Recently, Composite-Reinforced Mortars (CRM) and Fabric-Reinforced Cementitious Matrices (FRCM) systems allowed to overcome the undesirable increase in panel weight and stiffness of the traditional reinforce plaster and the chemical and physical compatibility issues with masonry substrates related to Fiber-Reinforced Polymer (FRP) stripes or sheets (Papanicolaou et al, 2008; Valluzzi et al., 2014). Furthermore, the need to comply with environmental requirements has led researchers to explore the potential of steel (Albanesi et al., 2023) and timber (Guerrini et al., 2021) exoskeletons as viable strengthening strategies. This paper presents a three-dimensional equivalent-frame macroelement developed to incorporate the biaxial contribution of lumped and distributed reinforcing and strengthening layouts into the static and dynamic response of masonry panels. The proposed formulation builds upon the macroelement proposed by Penna et al. (2014), which proved effective and efficient for the static and dynamic analyses of masonry structures within the equivalent-frame approach (Lagomarsino et al., 2013). However, despite the improvements carried out over the years (Bracchi et al., 2021; Bracchi and Penna, 2021), its formulation has always been restricted to the in-plane response of URM panels. In this work, an additional axial-flexural interface is provided at the center of the macroelement, allowing to reproduce the correct axial and flexural elastic without requiring manual adjustments to the mechanical properties or iterative algorithms (Bracchi et al., 2021). The biaxial axial-flexural response is obtained through either a stripe or fiber discretization of the interfaces, prioritizing computational efficiency on the one hand, as stresses are analytically integrated along each stripe (Penna et al., 2014), or constitutive law versatility on the other hand, as each fiber is assigned a uniaxial stress-strain relationship. Notably, the assembling algorithm responsible for the sectional integration proved suitable for explicitly introducing additional elements, such as lumped or distributed reinforcement, enforcing their collaboration with the macroelement interfaces via kinematic constraints. In previous studies, the proposed macroelement successfully reproduced the cyclic response of stone masonry piers strengthened with CRM applications (Salvatori et al., 2025a,b). In this paper, its capabilities are further investigated through the numerical simulation of quasi-static cyclic shear-compression tests conducted on two calcium silicate masonry piers. The first specimen is unstrengthened and serves as a reference, while the second is retrofitted with a timber exoskeleton and oriented strand boards (OSBs), mechanically connected to the masonry substrate (Guerrini et al., 2021). 2. Three-dimensional macroelement formulation Similarly to the formulation of Penna et al. (2014), the macroelement discussed in this paper consists of an assemblage of axial-flexural interfaces separated by shear-deformable central bodies (Fig. 1a). However, the proposed macroelement introduces a central interface in addition to the two located at the panel extremities (Vanin et al. 2020). This additional interface releases the kinematics of the macroelement and allows to natively capture the axial and flexural elastic stiffness of a masonry element without relying on manual adjustments or iterative corrections (Bracchi et al., 2021). Indeed, the three nonlinear interfaces serve as Gauss-Lobatto integration points, enabling the exact integration of a third-order polynomial expression, which corresponds to a second-order curvature profile. In this context, integration lengths of 1/6 and 2/3 the panel height are assigned to the end and central interfaces, respectively. The nonlinear biaxial axial-flexural response of the interfaces is computed using either a stripe (Fig. 1c) or fiber (Fig. 1d) discretization. In this context, a linear strain profile is enforced via kinematic constraints, as a function of the degrees of freedom associated to the interfaces (Fig. 1b).