PSI - Issue 78

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

1535

4. Conclusions This paper presents a novel three-dimensional equivalent-frame macroelement that extends the well-established two-dimensional formulation of Penna et al. (2014) to simulate the biaxial response of unstrengthened and strengthened masonry panels. More specifically, the out-of-plane and torsional degrees of freedom are incorporated, and an additional central interface is introduced to natively reproduce the correct axial-flexural elastic stiffness of the element. The biaxial axial-flexural response of the interfaces is computed according to a stripe or fiber discretization, favoring computational efficiency on the one hand, and constitutive model versatility on the other hand. Notably, the biaxial contribution of strengthening and reinforcement solutions can be explicitly modeled by adding analytical stripes or fibers with different mechanical properties to the macroelement interfaces, enforcing a linear strain profile through kinematic constraints to ensure proper collaboration. The proposed macroelement is validated against in-plane quasi-static shear-compression tests on two calcium silicate (CS) masonry piers with identical dimensions. The first pier is unstrengthened, while the second is retrofitted with a timber frame and OSB panels to enhance its flexural and shear performance, respectively. The numerical model of the bare pier shows good agreement with the experimental results, accurately capturing the initial stiffness, peak and residual strength, energy dissipation, and failure mechanisms. The choice of constitutive law for masonry in compression has irrelevant influence on the results, as the compressive strength is not reached during the applied displacement history. The model of the retrofitted pier shows only minor deviations from the experimental results in terms of lateral strength. However, the choice of constitutive law for the masonry influences the accuracy in predicting cyclic behavior and residual strength. While a parallel-elastic unloading branch improves the simulation of cyclic response, a bilinear model cannot capture the progressive strength degradation observed experimentally. Moreover, the multilinear model provides a closer match to the hysteretic response, accurately reproducing the lateral strength degradation up to the onset of the diagonal crack observed in the test. Overall, the numerical simulations confirm the proposed macroelement as a reliable and versatile tool for modeling unstrengthened and strengthened masonry elements, accommodating a wide range of retrofit solutions while offering different trade-offs between accuracy of results and computational efficiency. Future developments will incorporate the contribution of strengthening and reinforcing materials to shear behavior. Additionally, the macroelement will be further validated against experimental results from URM elements retrofitted with alternative techniques, such as CRM and FRCM systems. The proposed macroelement will subsequently be employed in the global modeling of entire masonry buildings to evaluate the effectiveness of various strengthening strategies in enhancing their seismic performance. Furthermore, thanks to its efficient formulation, the proposed macroelement will be used to investigate the effects of local or global strengthening interventions on URM building aggregates with reasonable computational effort. Acknowledgements This study is conducted within the DPC-ReLUIS (2024- 2026) Work Package 5 “Integrated and sustainable interventions for existing buildings” and Work Package 10 “Code contributions for existing masonry constructions” funded by the Italian Department of Civil Protection (DPC). The authors would like to thank Nederlandse Aardolie Maatschappij BV (NAM) and the EUCENTRE Foundation, Italy, for funding and coordinating the experimental program. Note that the opinions and conclusions presented by the authors do not necessarily reflect those of the funding entities. References Albanesi, L., Manzini, C. F., Morandi, P., 2023. Experimental In-Plane Seismic Performance of an Innovative Steel Modular Strengthening System for URM Walls. Journal of Earthquake Engineering, 28(5), 1331–1357. https://doi.org/10.1080/13632469.2023.2231087 Bracchi, S., Galasco, A., Penna, A., 2021. A novel macroelement model for the nonlinear analysis of masonry buildings. part 1: axial and flexural behavior. Earthquake Engng. Struct. Dyn. 50(8), 2233-2252. https://doi.org/10.1002/eqe.3445 Bracchi, S., Penna, A., 2021. A novel macroelement model for the nonlinear analysis of masonry buildings. part 2: shear behavior. Earthquake Engng. Struct. Dyn., 50(8), 2212-2232. https://doi.org/10.1002/eqe.3444