PSI - Issue 84

Luca Vené et al. / Procedia Structural Integrity 84 (2026) 544–551

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( , ) = ( , ) , +∑ ( , , ) ( ) =1

(6) Where ( , ) , is the displacement at the generic node k of the single pile as in the isolated case and ( , , ) is the soil flexibility coefficient relating to the displacement at the generic node k of the pile i to the load acting at the node l of the pile j . Then, the vertical equilibrium equations of the two piles are imposed as follows (Eq. 7), in which the value 1 represents the unit vertical loads applied at both the pile heads: ∑ ( ) =1 =1, ∑ ( ) =1 =1. (7) After assembling the compatibility conditions for all nodes of piles i and j , together with the equilibrium conditions, the complete coupled system can be resolved to assess the n i + n j + 2 unknows. Let w 1 , i and w 1,j denote the head settlement of pile i and pile j under unit vertical load, respectively, in the isolated case and let w i,pair be the head settlement of the pile i extracted from the solution of the pair of piles analysis described above. Obviously, w 1 , i and w 1 , j are equal to 1/K s,i and 1/K s,j , respectively. The interaction coefficient α ij (Poulos, 1968; Poulos and Davis, 1980) is defined as shown in the following Eq. 8: = ,pair − 1, 1, . (8) The pile-pile interaction coefficient quantifies the influence of the axial load on pile j on the axial displacement of pile i (e.g.). By superposing the effects of all neighbouring piles, the vertical settlement of pile i within the group ( w i,group ) is obtained as shown in the following Eq. 9, in which w i is the settlement of pile i in the isolated case due to its own load Q i (Eq. 1). , = +∑ 1, =≠1 (9) 2.3. Computation of Performance-Based Interaction Domains To derive the interaction domain of a pile group in the Q–Mx (or Q-My) space, the analysis is performed by prescribing limit boundary conditions at the pile cap. Each imposed roto-translational state represents a limit configuration of the foundation response, and the corresponding pile-head displacements are computed accordingly. The pile cap is assumed to be perfectly rigid, with piles connected through spherical hinges, so that the kinematics reduce to a rigid-body roto-translation about the cap centroid. Within this framework, pile-to-pile variability in mechanical and geometric properties can be explicitly accounted for, provided that the pile-group layout remains symmetric with respect to the principal axes, ensuring consistency of the imposed rigid-body motion. The displacement field of the rigid pile cap is fully described by two generalized kinematic parameters: the vertical settlement at the cap centroid, denoted as 0 , and the rotation about the -axis (or y -axis), denoted as . For each pile , the quantity represents the horizontal distance of the pile head from the rotation axis lying in the plane of the pile cap. For a given pair ( 0 , ) , the target vertical displacement at the head of pile is therefore obtained from Eq. 10: