PSI - Issue 80

Riccardo Giacometti et al. / Procedia Structural Integrity 80 (2026) 219–231

221

R. Giacometti, N. Grillanda, V. Mallardo / Structural Integrity Procedia 00 (2023) 000–000

3

interaction in two-dimensional (2D) masonry structures was presented: masonry was here modelled as rigid blocks (the voussoir) connected by unilateral springs, whereas the soil was represented as an elastic half plane and modelled via the Boundary Element Method (BEM). The second part of the presented research is to extend the procedure to the general three-dimensional (3D) case.

2. Seismic analysis

The analysis of existing masonry structures under actions induced by seismic events is typically conducted by applying a configuration of horizontal load proportional to self-weight. The technical and scientific literature on nu merical methods for such problems is abundant, see Occhipinti et al. (2022) or Cannizzaro et al. (2022). In the case of existing masonry structures, where the material mainly behaves as an assembly of rigid blocks connected by fric tional or unilateral interfaces, limit analysis is one of the most e ff ective tool for the computation of the horizontal load-bearing capacity.

2.1. Limit analysis

Several numerical methods for the limit analysis of masonry structures have been presented. One of the most adopted strategy is the rigid block limit analysis, in which each brick or stone is modelled as a rigid and infinitely resistant block whereas mortar joints are simplified as zero-thickness frictional interfaces. Such a representation is not far from the reality, since mortar is often degradated or absent in historical masonry, thus resulting in collapse modes based on rigid blocks mechanisms. Basing on such assumptions, it is quite easy to express the two limit analysis theorems as linear programming problems and to implement simple codes aimed at determining collapse behaviour and load-bearing capacity of rigid blocks structures (see also some previous publications from some of the authors Tiberti et al. (2020), Grillanda et al. (2022)). However, it should be mentioned that rigid block limit analysis can be applied provided that a detailed physical modelling of the structure, with representation of the actual texture, can be realized. In addition, structures composed of a huge number of bricks can lead to models with a lot of elements in which the derived computational cost may compromise the reliability of the numerical analysis.Thereofre, alternative limit analysis methods have been recently presented. The approach adopted here relies on a very simple modelling strategy based on the definition of a homog enized material which maintains a collapse behaviour equivalent to the heterogeneous masonry. A given masonry structure is here discretized into a homogenous mesh of triangular elements, independently of the actual masonry texture. Each triangular element is assumed rigid-plastic, thus its kinematics is governed by the rigid velocities of its centroids and the plastic strain rates associated with the element (Fig. 1). In two dimensions (2D), the kinematics of any node P belonging to the i-th element is expressed as follows: where the vectors x P and x i are the spatial coordinates of node P and centroid of the i-th element respectively, v P and v i are the corresponding translational velocity vectors, Φ i is the skew-symmetric rotation tensor defined from the in-plane rigid rotation rate ϕ i of the element, and E i is the plastic strain rate tensor containing the in-plane plastic strain rates ϵ i 11 , ϵ i 22 , ϵ i 12 . The centroid velocities, rotations and plastic strain rates are unknown in the kinematic problem. Note that in this model the plastic strain rates are representative of damage in the masonry structure, thus no velocity jumps are allowed between adjacent elements. To exclude velocity jumps, a nodal compatibility condition must hold: this means that, if P is a common node between multiple elements, the quantity v P is the same whether of these elemens is considered for the computation of Eq. (1). We consider now a distribution of loads subdivided into dead loads f D and live loads λ f L , in which λ is a load factor a ff ecting the live loads only. In particular, f L consists of a horizontal load equal to self-weight. It is possible now to write a simple linear programming (LP) problem on the basis of the kinematic theorem of limit analysis. For v P = v i + Φ i x P − x i + E i x P − x i (1)