PSI - Issue 84

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

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1. Introduction The management and maintenance of road and railway bridges increasingly rely on continuous structural monitoring to guarantee safety, durability, and serviceability throughout their operational life. This necessity is strongly linked to the steady growth of freight transport and traffic volumes, which results in a significant increase in load cycles acting on bridge structures and, consequently, in a higher vulnerability to fatigue-related damage if not properly controlled (Frýba, 1999; Chopra, 2012). From a dynamic perspective, moving loads play a central role, as they can substantially amplify structural responses and activate complex interaction mechanisms between vehicles and bridges (Green and Cebon, 1997; Yang et al., 2004). The direct dynamic problem of bridges subjected to moving loads has been widely investigated over the past decades using both analytical and numerical approaches. Early studies modeled bridges as beams crossed by moving point loads, allowing the identification of resonance conditions and dynamic amplification factors (Frýba, 1999). These formulations were later extended to more refined Vehicle–Bridge Interaction (VBI) models, which account for the distributed mass and dynamic properties of vehicles (Green and Cebon, 1997; Yang et al., 2004). Comprehensive overviews of VBI modeling and advanced simulation frameworks confirm the maturity reached in the analysis of direct moving load problems (Law and Zhu, 2004; Han et al., 2024). In contrast, the inverse problem of Moving Load Identification (MLI) has gained increasing attention only in more recent years, particularly for road bridges, where traffic characteristics are highly variable and less predictable than in railway systems. The objective of MLI is to identify the number, magnitude, spacing, and sometimes eccentricity of vehicle axles by exploiting measured structural responses, such as displacements, strains, or accelerations. Conventional solutions often rely on invasive techniques, including weigh-in-motion systems or dedicated sensing devices installed on the roadway, which are accurate but costly and difficult to maintain (Yu et al., 2016). To overcome these limitations, response-based identification methods have been proposed, in which vehicle loads are inferred indirectly by solving an inverse dynamic problem using measured structural responses from pre-installed monitoring systems. Despite their potential, such inverse problems are inherently ill-posed and affected by significant uncertainties related to modeling assumptions, measurement noise, and damping characterization (Gattulli et al., 2019; Lofrano et al., 2016). In particular, damping represents a critical source of uncertainty, as it is strongly influenced by operational and environmental conditions and is difficult to estimate reliably in real structures (Gattulli et al., 2019). Several regularization strategies and optimization-based methods have been introduced to improve the stability and robustness of moving load identification. Approaches based on updated static components, influence lines, and advanced time-integration schemes have shown promising results in mitigating sensitivity to uncertain parameters (Mosleh et al., 2015). More recently, hybrid and data-driven techniques combining numerical modeling with signal processing and machine learning have demonstrated high accuracy in identifying vehicle weights, speeds, and road roughness conditions (Ghazal and Mwafy, 2023). Within the broader context of inverse structural problems, meta-heuristic optimization algorithms have become increasingly popular due to their ability to explore complex, multi-dimensional search spaces without requiring gradient information. Among these, Differential Evolution (DE) has emerged as a robust and flexible tool for parameter identification in structural dynamics. DE has been successfully applied to a variety of problems, including the identification of structural parameters in bridges and systems characterized by uncertainty or noisy measurements (Gattulli et al., 2019; Di Re et al., 2024). Its effectiveness has also been demonstrated in the context of MLI, where classical deterministic methods often fail due to nonlinearity and non-convexity of the objective functions. Despite these advances, the accuracy and computational efficiency of DE-based identification procedures strongly depend on the mechanical model adopted to describe the bridge response and on the definition of appropriate convergence criteria. Excessively simplified models may neglect relevant coupling effects, while fully three dimensional finite element models are often computationally prohibitive for iterative identification procedures. In this respect, reduced-order analytical models represent a suitable compromise between accuracy and efficiency (Lofrano et al., 2016). In this framework, the present work focuses on the identification of travelling loads acting on skew road bridges by combining a reduced-order analytical model with a Differential Evolution algorithm. The bridge is modeled as a one-dimensional Euler–Bernoulli beam with bending–torsional coupling, a formulation capable of capturing the essential dynamic features of skew bridges while maintaining low computational cost. This modeling choice allows