PSI - Issue 44

Micaela Mercuri et al. / Procedia Structural Integrity 44 (2023) 1276–1283 Author name / Structural Integrity Procedia 00 (2022) 000–000

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1. Introduction Recent earthquakes showed that vaulted structures are prone to show extensive damage conditions and local or global collapse when subjected to the seismic action (Lagomarsino (2012), D'Altri et al. (2017), Clementi (2021)). Recently, an increasing attention is paid by researchers on this topic, that on one side aim to a better understanding of the unreinforced masonries and, on the other hand, strive to find performing strengthening systems (Vailati et al. (2021)). Aiming to assess the safety of a vaulted system, one should analyze two different structural elements, i.e. thrusting arches or vaults and the supporting vertical panels that resist the thrusts (Block et al. (2006)). These latter masonry elements are simultaneously subjected to in-plane (Gregori et al. (2022)) and out-of-plane actions under the seismic excitation (Najafgholipour et al. (2013)) and, particularly, the out-of-plane load provokes a mechanism of progressive overturning of the buttresses (Mercuri et al. (2020) and Mercuri et al. (2021)). As a consequence, the above-vaulted structure first tries to accommodate the imposed movements through the appearance of a diffuse fracturing process and, as the spread of the abutments keeps increasing, a kinematic mechanism triggers, and a progressive condition of instability is reached (Mercuri et al. (2022)). Nowadays, two are the main existing methods for the analysis of masonry arches and vaults: i.e. elastic analysis and limit analysis (Onat and Prager (1953)). The elastic analysis of masonry structures, introduced by Navier in 1926, provides results related to the stress state in a statically indeterminate structure, assuming that the material is a continuum linear-elastic solid. Despite the possibility to reach a closed form solution, the method is inappropriate in analyzing masonry vaulted structures because of three main reasons (Ochsendorf (2002)): (i) the overall non-linear structural deformations are not only due to the elastic deformation of the masonry material; (ii) the exact stress state is unknown in a masonry structure since the exact loading history, the boundary conditions and the material proprieties are affected by high levels of uncertainty and randomness; (iii) the consideration of the masonry as an elastic continuum material is an approximation that in most of cases is not realistic. Limit analysis has been progressively used to overcome the listed drawbacks related to elastic analysis. The assumptions underlined to limit analysis provide no-tensile strength, an infinitive compressive strength and the negligence of sliding between blocks (Heyman (1969)). While performing the limit analysis, the spread increasing of the buttresses causes the thrust line to reach the edge of the arched structure and, instantly, a hinge develops and the entire section cracks. The formation of a number of hinges, each one being caused by the thrust line becoming tangent to the boundaries of the arch causes a kinematic mechanism, which leads to the collapse state. Limit analysis is considered currently as an effective tool for understanding the behavior of unreinforced masonry structures, especially for the analysis of arches and vaults because it directly identifies the structural ultimate conditions and allows a fast and reliable control of safety levels (Milani et al. (2008), D’Ayala et al. (2008)). However, limit analysis is affected by three main limitations: (i) it completely ignores the damage propagation phenomenon, starting from the trigger of the fracture up to the complete structural failure; (ii) limit analysis is performed under the hypothesis that the material has no tensile strength and it was proven that for quasi brittle materials, the no-tension design is not guaranteed to be on the safe side (Bazant (1996)); (iii) limit analysis does not capture size effect (Bazant and Lewis (2003) and Mercuri et al. (2022)). To overcome the listed limitations, one should apply a computational strategy that resorts on non-linear fracture mechanics concepts (Bazant et al. (2021)). There are several mathematical and numerical methods able to simulate the fracturing behavior of quasi-brittle materials with different degrees of complexity. For example, meso-scale approaches simulate directly the meso-structure and allow an accurate modeling of the phenomena undergoing in the fracture process zone in both 2D (Carol et al. (2001)) and 3D (Caballero (2006)). Other micro-scale approaches also exist, but they are in general more expensive computationally (Drougkas et al. (2014), Gregori et al (2020) and (2021)). The choice of the most appropriate strategy should balance the desired levels of accuracy and the computational costs. In this study, the Lattice Discrete Particle Model (LDPM) (Cusatis et al. (2011) and (2011b)) is adopted. The model is able to capture the mechanical behavior of granular quasi-brittle materials at the meso-scale, it simulates the heterogeneity of the material at the particle level (Yang et al. (2021, 2022), Feng et al. (2022), Cibelli et al. (2022)) and it is suitable to tackle the behavior of unreinforced masonries (Mercuri et al. (2021b) and Angiolilli et al. (2020)). The interaction between particles is governed by three sets of constitutive equations describing tensile fracturing with strain softening, cohesive and frictional shearing, and compressive response with strain-hardening (Schauffert et al. (2012)). Through the adoption of LDPM, this paper proposes to overcome the lack of in-depth analysis of arches and vaults under spreading supports. The non simultaneous formation of hinges, the activation of the mechanism and the