PSI - Issue 84

Andrea Mileto et al. / Procedia Structural Integrity 84 (2026) 829–836

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the identification of not only the magnitude and number of axle loads, but also their spacing and transverse eccentricity, which are particularly relevant for skewed configurations. The model and the identification technique have been presented by some of the Authors in a recent paper (Mileto et al. 2026). A key contribution of the proposed methodology lies in the adoption of a modified convergence criterion for the DE algorithm, referred to as a swarm-based criterion. Unlike traditional tolerance-based stopping conditions, this criterion enforces proximity among all individuals in the population, thereby reducing the risk of premature convergence toward local minima. Although this approach may increase the number of generations required, it enhances the robustness and reliability of the identification process. Here the method is improved and validated considering more generalized conditions, then testing its potential validity and reliability in real-life situations. 2. Moving Load Identification technique This section sum up the adopted Moving Load Identification technique; for a full description see the recent paper published by some of the Authors (Mileto et al., 2026). 2.1. Bridge model This section describes the mechanical model adopted to represent the dynamic response of a skew bridge subjected to travelling loads. The bridge is modeled as a reduced-order, one-dimensional Euler-Bernoulli beam with bending torsional coupling, an approach that allows capturing the essential dynamic behavior of skew bridges while maintaining a limited computational cost. The model neglects axial deformability and transverse shear effects, focusing instead on two primary kinematic variables: the vertical displacement and the torsional rotation of the bridge cross-section. These two fields are coupled through the skewness angle, which represents the inclination between the bridge longitudinal axis and the support lines and plays a key role in activating torsional responses even when loadings are not eccentric. The governing equations of motion are formulated by considering the combined effects of inertia, stiffness, and damping in both bending and torsion, together with the external forces and moments induced by moving vehicle loads. Travelling loads are modeled as concentrated forces acting at prescribed distances, moving along the bridge span, with possible transverse eccentricity with respect to the bridge centerline. This formulation enables the representation of vehicles with multiple axles, different load intensities, and non-uniform axle spacing. To solve the resulting partial differential equations, a Faedo–Galerkin discretization is employed. The vertical displacement and torsional rotation fields are approximated as linear combinations of a finite number of admissible mode shapes, leading to a system of ordinary differential equations expressed in terms of generalized coordinates. The number of retained modes is selected through a convergence analysis based on the stabilization of peak accelerations, which are one the key response quantities exploited in the identification procedure. This choice ensures that the model accurately reproduces the relevant dynamic features of the bridge without introducing unnecessary computational complexity. 2.2. Identification scheme The moving load identification methodology is formulated as an inverse problem solved through a Differential Evolution (DE) optimization algorithm, initially introduced by Storn and Price (Storn and Price, 1997). The objective of the identification process is to determine unknown load parameters, including axle magnitudes, spacing, and transverse eccentricity, by minimizing the discrepancy between measured and simulated bridge responses. The comparison is performed in terms of acceleration signals recorded at selected monitoring points along the bridge, exploiting target quantities in both frequency and time domain. After preliminary analyses (Mileto et al., 2026), where several objective functions were compared, two of them are here considered: =√ 2 + 2 + 2 , =√ 2 + 2 + 2 + 2 (1)