A 3D macroelement formulation for modeling unstrengthened and strengthened masonry piers

The widespread presence and seismic vulnerability of unreinforced masonry (URM) structures have prompted extensive research into their assessment and retrofit. Even when local vulnerabilities are mitigated and a global three-dimensional behavior is promoted, the seismic performance under horizontal excitations might still be insufficient. To address this limitation, strengthening interventions using composite materials, such as Fabric-Reinforced Cementitious Matrices (FRCM) or Composite-Reinforced Mortars (CRM), as well as retrofitting systems based on steel or timber exoskeletons, are often applied to one or both sides of the masonry walls. This paper introduces a novel three-dimensional macroelement developed to effectively capture the coupled in-plane and out-of-plane response of unstrengthened and strengthened masonry elements, balancing computational efficiency with constitutive law versatility through a stripe or fiber discretization of the cross-section. Indeed, the proposed formulation facilitates the introduction of additional reinforcement, enabling the explicit modeling of a wide range of strengthening solutions. The effectiveness of the proposed macroelement in reproducing the lateral response of masonry piers are proved through the numerical simulation of experimental quasi-static cyclic shear-compression tests.