PSI - Issue 78

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

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Fig. 1. (a) Local degrees of freedom of the three-dimensional macroelement, (b) interface degrees of freedom, (c) stripe and (d) fiber discretization of the cross-section.

In the first case, the interfaces are subdivided into analytical two-dimensional stripes, and the biaxial response is obtained by numerically integrating the contribution of each stripe along the thickness of the macroelement. As the in-plane response of each stripe is computed through the analytical integration of stresses proposed by Penna et al. (2014), it results in a computational efficient formulation. In this context, a bilinear material model with recentering unloading is adopted in compression, while an elasto-fragile relationship is assumed in tension (Fig. 2a). Alternatively, the interfaces are subdivided in a series of fibers, whose nonlinear contribution is numerically integrated along the two principal directions to obtain the three-dimensional response of the cross-section. As each fiber is assigned a uniaxial stress-strain relationship, more detailed material models can be adopted. In this context, the constitutive law proposed by Bracchi et al. (2021) (Fig. 2b) and the multilinear model of Fig. 2c are provided. The shear and torsional responses of the macroelement are concentrated in the central bodies, and are considered independent from other actions and along orthogonal directions. As in the formulation of Penna et al. (2014), the shear behavior is based on the continuum damage model for masonry developed by Gambarotta and Lagomarsino (1997a,b), which has been macroscopically integrated to align with the macroelement formulation (Penna et al., 2014). On the other hand, the torsional response is maintained linear elastic, as typically of minor concern for masonry elements.

Fig. 2. Interface constitutive laws: (a) Penna et al. (2014), (b) Bracchi et al. (2021), and (c) multilinear material models.