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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8

4. Do until convergence ( i ≥ 1): (a) Relocate u s

i as given boundary conditions u m

i on the masonry base

(b) Compute base reactions r m

i of the loaded masonry structure with u m

i as b.c.

i on the elastic half space as t s i

(c) Relocate r m

(d) Compute displacements u s

i + 1 on ¯ S of the elastic half-space under t s i

(e) Check errors

At each iteration step the convergence is improved by applying an extension of the technique proposed by Elleithy and Tanaka (2003), that is by imposing the following boundary conditions at steps (4b) and (4d): u m i = (1 − α ) u m i − 1 + α u m i (6a) t s i = (1 − β ) t s i − 1 + β t s i (6b) where α and β can be optimized by a preliminary parametric analysis. In the numerical example presented further α = β = 0 . 2 is adopted. The iteration process is halved provided that some tolerances are satisfied w.r.t. the normalised root mean square errors. Steps 4(a) and 4(c) require attention. They are better detailed in the next two subsections as they depend on the modelling approach. The mechanical model of the masonry is similar to what described in Sect. 2.1. The brick is still a rigid (with infinite compressive strength) 3D cell in unilateral contact (and with finite Mohr-Coulomb friction capacity) with the surrounding bricks. On the other hand, the contact is modelled by discrete springs in the normal and in the tangential directions that are located in the corner points of each brick-brick contact area. In other words, the elasticity of brick and mortar is lumped with the springs. The response in terms of contact forces (including r m of step 4b) under load is obtained (see also Mallardo and Iannuzzo (2025)) by minimizing the Total Complementarity Energy ( t means transpose): 3.2. Masonry modelling

1 2

F t AF − BF t ¯ U

TCE =+

(7)

under the constraints:

CF + b + Dq = 0

(8a) (8b)

| f k

k N

T |≤ µ f

for k = 1 , · · · , n nodes

where F collects the internal nodal contact forces f N , f T 1 , f T 2 , BF t is the boundary emerging force vector, ¯ U the displacement field imposed at the base, b collect the volume forces lumped in with the brick’s centroid, q collects possible additional concentrated loads relocated in the brick’s centroid by the operator D , C is the equilibrium matrix. The resolution of the minimization problem provides the base reactions r m at step 4(b) of each iteration. Such reactions are point-concentrated forces whereas the soil model (see next subsection) requires tractions (i.e. force per unit area), that is step 4c. This is simply obtained by spreading r m uniformly, that is (see Fig. 5): t s = − r m d 1 d 2 , (9)

3.3. Half-space modelling

The soil problem of step 4(d) is solved by applying a typical integral equation approach. Such an approach is governed by an integral equation involving displacements and tractions on the boundary ¯ S of the domain Ω under