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

228 10

i i φ δ

u

Fig. 6: Interaction between the rigid body motion of the masonry cell and the soil nodes.

The inversion of Eq. (12) provides the solution recalled in the iteration scheme at steps 3 and 4(d): u s = H − 1 b

(15)

Such a solution is obtained in terms of three components for each node of the surface ¯ S of the soil. It needs to be relocate to rigid motion of each brick as required by step 4(a). To do so a more dense layer of cells is inserted between bricks and soil. If (see Fig. 6) : u m i = ( δ 1 ,δ 2 ,δ 3 ,ϕ 1 ,ϕ 2 ,ϕ 3 ) (16) describes the rigid kinematics of the cell ( δ k being the displacement component in the x k direction and ϕ k the rotation around x k ), the relocation can be obtained by solving in the least square sense the following overdetermined system of equations: u m = A − 1 u s (17) where: A =     I 3 × 3 skw( x G − x l ) I 3 × 3 skw( x G − x l + 1 ) I 3 × 3 skw( x G − x l + 2 ) I 3 × 3 skw( x G − x l + 3 )     (18) inwhich I 3 × 3 is a 3 × 3 identity matrix, skw() is the operator that forms the skew-symmetric matrix associated with the indicated vector, x G and x l ,..., l + 3 are the vectors containing the coordinates of the brick centre of gravity and the nodes l , . . . , l + 3 of the element l . One numerical example is presented in this subsection as more numerical analyses are still in progress. It refers to a typical two span river bridge as scouring may be well simulated by the proposed approach. The masonry is loaded by self-weight. In Fig. 7 the masonry discretization is depicted. The backfill is modelled as weaker ( ϕ = 15 rather than 40, k = 10 9 rather than 10 11 ) with respect to the arch. In Fig. 7 the thrust line (in green) and the crack opening framework (in red) are depicted assuming fixed foundation (classical approach). Thrust line and cracks result to be symmetrical as they cannot be influenced by the soil deformation. In Fig. 8 are depicted the results obtained by the proposed approach. The influence of the soil (Young’s modulus E = 50 MPa , Poisson’s coe ffi cient ν = 0 . 25) is highlighted in the figure by the shifting of the cracks that, of course, are still symmetric as the action is symmetric. A di ff erential settlement of 1 . 3cm is computed between the central pier and the lateral ones that could not be estimated with the classical approach. 3.4. Numerical example