PSI - Issue 84

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

832

where E values are the square roots of normalized discrepancies among experimental data and those obtained from the beam model varying the load parameters. More specifically: E Q compares the peaks in frequency domain (weighted on their maximum value and on the sequential order), E G the effective accelerations, E t the total vehicle crossing time, E a the maximum and minimum accelerations. The Differential Evolution algorithm is used to explore the multidimensional parameter space associated with the unknown load characteristics. DE is particularly suited for this problem due to its ability to handle nonlinearity, multimodality, and noisy objective functions without requiring gradient information. A key innovation introduced in (Mileto et al., 2026) is a swarm-based convergence criterion, which replaces traditional tolerance-based stopping conditions. In few words, instead of terminating the optimization when a single best solution satisfies a predefined threshold, the algorithm continues until all individuals in the population converge toward sufficiently close solutions. This approach significantly reduces the risk of premature convergence to local minima and improves the reliability of the identified parameters. 3. Improvement and validation The improvement and validation of the proposed technique is performed through several numerical simulations assuming the following beam parameters: • Bridge length and width = 20 m, = 5 m • Skewness angle = 0° or 18° (both the non-skew and the skew cases are considered) • Mass per unit length = 4648.6 kg/m • Mass moment per unit length = 8497.5 kg m • Damping ratio =0.01 • Bending stiffness = 1.19 ∙ 10 10 Nm 2 • Torsional stiffness = 4 ∙ 10 9 Nm 2 The vehicle is a two-axle vehicle (a two-load dynamic force system in the adopted model), whose parameters are: • Speed = 10 m/s • Axle loadings 1 =40kN, 2 =80kN • Axle spacing Δ=4.8m • Load eccentricity = 1 m In the analysis, assuming known the beam properties, the DE is adopted to identify: the load magnitudes corresponding to the forces associated with the vehicle axles, respectively; the axle spacing Δ ; and the eccentricity . When simultaneous variations of multiple parameters are considered, for instance by varying 1 and at the same time, the problem becomes more complex, effectively resulting in a multi-dimensional search problem. To validate how the chosen objective functions manage this complexity, sensitivity analyses are performed. In detail: the load magnitudes 1 and 2 are combined imposing a constant load difference ( Δ = 2 − 1 = const) and a constant total load ( TOT = 1 + 2 = const). The two selected objective functions (Eqs. (1)) are then evaluated under both configurations. A third case is also considered, adopting a reduced number of modes in the reconstruction of numerical solution. 3.1. Case 1 ( = const ) Table 1 shows the computed load simulations, where = 1 0 / 1 , with 1 0 =40kN the initial considered value. The results are showed in Fig. 1 in terms of: the horizontal axes is (basically, the load perturbation), the vertical one is the multiplier imposed to perturb the eccentricity b , and the colourmap represents the value of the objective functions when perturbed values are compared with unperturbed ones (values when abscissa and ordinate are both zero). These surfaces yield a distribution characterized by a certain inclination, which can be interpreted by considering the equivalence between different combinations of load magnitude and eccentricity: a smaller load acting with a larger