PSI - Issue 80

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

227

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

9

r

N T1 t , t , t

T2

d

1

d

2

Fig. 5: Masonry reactions - soil traction interaction.

analysis:

c i j ( x s ) u i j ( x s , x f ) + − S

T i j ( x s , x f ) u j ( x f ) dS ( x f ) = S

U i j ( x s , x f ) t j ( x f ) dS ( x f )

(10)

where S is the free surface in our case, x s and x f are called source and field point, respectively, c i j = δ i j / 2 ( δ i j being the delta Kronecker) on smooth surfaces and − stands for Cauchy principal value. Matrices U i j and T i j collect the so-called fundamental solutions that depend on the type of problem under analysis. For the problem under analysis the Mindlin solution, that is the solution in a half-space due to a point force, is used. Its expression can be found in Brebbia et al. (1984), Aliabadi (2002). With such a choice, ¯ S coincides with the free surface where it occurs that T i j = 0, and the solution of Eq. (10) automatically includes the semi-infinite size of the domain. T i j = 0 on the free surface means that the discretization can be carried out only on the part of the free surface that is loaded, hence, in the problem under analysis, it is limited to contact area ¯ S . Both properties represent a huge advantage w.r.t. a Finite Element approach. Linear quadrilateral shape functions M n ( ξ ) (of Jacobian J ( ξ )) are adopted on ¯ S to represent both the geometry and the displacement,traction fields. Such a choice is supported by the need to avoid detached subzones inside the element. After discretization and collocation of x s in x s , the Eq. (10) can be written as:

j

+ 1 − 1

4 n = 1

EL l = 1

+ 1

u n

l ( ξ ) d ξ

c i j ( x s ) u j ( x s ) +

T i j ( x s , x f ( ξ )) M n ( ξ ) J

=

− 1

j

+ 1 − 1

4 n = 1

EL l = 1

+ 1

t n

l ( ξ ) d ξ

U i j ( x s , x f ( ξ )) M n ( ξ ) J

(11)

− 1

Eq. (11) written in each node of ¯ S provides the following final governing system of equations: Hu s = b whose solution provides u s of steps 3 and 4(d). The singularity in the terms of the matrix H , of the type: + 1 − 1 + 1 − 1 T i j ( x s , x f ( ξ )) M n ( ξ ) J l ( ξ ) d ξ is strong and it is sorted out by the well-known rigid body condition Brebbia et al. (1984). The right hand side b collects terms of the type: (12) (13)

j

+ 1 − 1

+ 1

t s

l ( ξ ) d ξ

U i j ( x s , x f ( ξ )) M n ( ξ ) J

(14)

− 1

t s j is a component of t s . The singularity of the involved integral is weak and it is solved by a subdivision approach with mapping (see Gao and Davies (2011)) that reduces the order of singularity and makes the Gaussian quadrature reliable. For a source point close (but not inside) the element, both integrals are evaluated by a subelements approach that divides the original element into smaller elements (closer to the source point) in order to reach a reduced error with a pre-set number of Gaussian points.