PSI - Issue 84

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

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Assuming elastic behaviour and perfect bonding at the pile–soil interface, i.e. in the absence of relative slip, displacement compatibility between the pile and the surrounding soil is enforced at each computational node (Eq. 2). Accordingly, for each node i =1,2,…, n v , the vertical displacement of the pile, w i,p , is equal to the corresponding soil displacement, w i,s . The soil displacement , at each node is obtained using the Mindlin solution for a concentrated load acting within an elastic half-space. The pile displacement , is calculated based on elastic theory, taking into account the distribution of axial force along the pile from the head to the -th node, as described in Viggiani (1999), assuming that the shear stresses at the pile–soil interface remain constant along each cylindrical segment of the discretized pile. , = , (2) The n v compatibility equations must be accompanied by a vertical equilibrium condition (Eq. 3) ensuring that the sum of q i and p equals the applied vertical load Q : ∑ =1 + = (3) The system of n v + 1 equations with n v + 1 unknows (i.e., the total pile head settlement w , the q i resultants of the n uniform shear stresses along the pile shaft and the p resultant of the base uniform stress) is then resolved, under linear elastic hypothesis. The initial axial stiffness of the pile is obtained from the computed settlement, w 1 , under unit axial load, i.e. Q = 1 (Eq. 4). = 1 1 (4) 2.2. Modelling of Pile–Pile Interaction Effects When two piles are embedded in the same elastic half-space, the load applied to one pile induces vertical displacements at the nodes of the adjacent pile, resulting in additional head settlements compared to the isolated case. This interaction is accounted for by extending the compatibility and equilibrium conditions to the coupled response of both piles. Let piles i and j be discretized into n i and n j vertical computation nodes, respectively. For each node k of pile i and each node l of pile j , the compatibility between the vertical displacement of the pile and that of the soil reads as shown in Eq. 5: ( , ) = ( , ) , = 1, 2, … , . (5) ( , ) = ( , ) , = 1, 2, … , As in the single-pile analysis, the vertical displacement of the piles is evaluated by enforcing compatibility between pile and soil displacements at each computational node. The same procedure described for the isolated pile is extended to the pair of piles, while accounting for the additional displacements induced by the neighbouring pile through pile– pile interaction. The soil displacement at node k of pile i is obtained by superposition of self-interaction and pile-pile interaction induced by pile j as observed in Eq. 6.