PSI - Issue 84

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

549

target , = 0 − ,

(10) The set of limit configurations is generated using the rectangular frontier method, whereby the boundary of a rectangular domain in the ( 0 , ) plane is sampled to identify extreme admissible rigid-body motions of the pile group. When multiple piles are present, the settlement of each pile is influenced not only by its own applied load but also by the loads carried by the other piles in the group through pile–pile interaction. To account for these effects, the hyperbolic load–settlement response of an isolated pile is corrected by superposing the interaction contributions from the surrounding piles. For the generic pile , the resulting load–settlement relationship is expressed as (Eq. 11): , = 1 , lim , lim , − +∑ 1 , =≠1 (11) where the first term represents the nonlinear settlement of the single isolated pile, while the second term accounts for pile–pile interaction effects induced by the loads applied to the other piles in the group. For each loading configuration imposed at the pile cap, defined in the ( 0 , ) plane, the target settlement w target,i is prescribed for each pile. The difference between the computed and target settlements defines the residual associated with pile : = [ 1 , lim , lim , − +∑ 1 , =≠1 ]− target , (12) The problem is then formulated as a bounded nonlinear least-squares optimization, in which the vector of pile loads is determined by minimizing the sum of the squared residuals (Eq. 13): ∑ 2 ( ) =1 (13) subject to the axial capacity constraints in Eq. 14: ( min) < < ( max) , = 1, … , , (14) where the bounds represent the tensile and compressive capacities of the generic pile i (Eq. 15): ( max) = , , ( min) =− ( ) , . (15) The nonlinear system is solved using MATLAB’s built-in function lsqnonlin (The MathWorks Inc., 2025), which implements a trust-region–reflective algorithm specifically suited for bounded nonlinear least-squares problems. The algorithm iteratively updates the vector of unknown pile loads so as to minimize the residual norm while ensuring that the solution remains within the physically admissible domain defined by the tensile and compressive capacity limits of the piles.