PSI - Issue 80

Riccardo Giacometti et al. / Procedia Structural Integrity 80 (2026) 219–231 R. Giacometti, N. Grillanda, V. Mallardo / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 1: Rigid plastic kinematics for a generic triangular element: initial configuration in grey, configuration after rigid body motion in blue, config uration after plastic deformation in red.

2D configurations, and in the case of dry-joint masonry (typical of historic constructions), the problem becomes:

minimize λ = − f T subject to C v T ϕ T ϵ T T D v

T ϕ T ϵ T = 0

T

(2a)

(2b)

A fix v P ϵ T ω T

T ϕ T ϵ T T

T

= 0

(2c)

≤ 0

(2d)

L v

T ϕ T ϵ T

T

f T

= 1

(2e)

where (2a) is the objective function, which derives from the application of the Principle of Virtual Power (note that no internal power is present when dry-joints are considered), (2b) is the nodal compatibility condition for the global model, (2c) is the geometric constrains, (2d) is the plastic flow rule, and finally (2e) is the normalization of the live load power. Some words on the definition of matrix P , which expresses the associated flow rule for the homogenized material in matrix form, will be spent in the next subsection. In the classical rigid blocks limit analysis, i.e. without homogenization, relations (2b) is absent and (2d) is replaced by a standard Mohr-Coulomb friction condition. The LP problem (2) returns the rigid velocities v , the rigid in-plane rotations ϕ , and with homogenization the plastic strain rates ϵ and the micro-rotations ω . Such quantities identify a mechanism. Note that the micro-rotation ω is not involved in the element kinematics (Eq. (1)) but it consists of an additional degree of freedom required to express the associated flow rule: we refer to Grillanda and Mallardo (2025) for a detailed discussion on this aspect. In addition, Eq. (2) allows to obtain a load factor associated to the mechanism. It must be observed that in such a model the result is a ff ected by the adopted discretization, since the strains are assumed constant within the element. Such an issue can be overcome by applying an iterative local mesh refinement on the element subjected to plastic strains. The homogenization theory is here followed for the definition of an homogeneous material equivalent to the hetero geneous masonry. The equivalence is here expressed through a procedure of compatible identification: first, the plastic strain rates in the homogeneous material must be representative of the kinematics of the heterogeneous material and texture with respect of the associated flow rule; second, the internal power per unit area must be the same between the two materials. Since in dry-joint masonry the internal power is null, we focus here on the first condition. Such a condition will be imposed via the matrix P within the LP problem (Eq. 2d). A reference volume element (RVE) is identified for the heterogeneous material. By assuming each brick as a rigid and infinitely resistant block, the RVE consists of four adjacent bricks, see Fig. 2(a), as in the classic work by de Buhan and de Felice (1997). The kinematics of any point P belonging to a rigid block having centroid G can be expressed 2.2. Homogenization