PSI - Issue 84

Marco Bonopera et al. / Procedia Structural Integrity 84 (2026) 457–464

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1. Introduction The advantages of prestressing the concrete were demonstrated over time by many civil engineering applications. Thus, large facilities were constructed using this technology, such as both long-span bridges and viaducts. Therefore, evaluating the residual load-carrying capacity of aging bridges became a significant task (Saudi et al. 2000). Experiments are the most straightaway approach to study both flexural and shear behavior of these structures. Some works investigated the flexural behavior of full-scale Prestressed Concrete (PC) girder-bridges (Botte et al. 2021; Wang et al. 2021; Kralovanec et al. 2022), whilst others focused on their shear behavior (Oh and Kim 2004; Huber et al. 2018; Lantsoght et al. 2021). These studies proved how existing PC girder-bridges with none-to-slight damage or deterioration ensured satisfactory load-bearing capacity, overperforming structural requirements by design codes. Nonetheless, tests on full-scale PC girder-specimens could be highly demanding (Bagge et al. 2018). This being the reason why reduced-scale PC girder-bridges were usually studied (Murray et al. 2019; Belletti et al. 2020). E.g., vibration measurements were useful applications to estimate their flexural rigidity. Hamed and Frostig (2006), Limongelli et al. (2016) and Bonopera et al. (2019) found that the stiffness of PC girder-bridges, with a parabolic or a straight tendon, only significantly varies under the effect of crack initiation or re-opening. Yet, Noble et al. (2016) executed experiments on a number of reduced-scale post–tensioned concrete girder-bridges. Such researchers showed that the slight decrease in fundamental frequency with increasing prestressing is not related to the softening effect. Thus, they concluded that the compression-softening theory must be deleted from consideration of all prestressed girder-bridges. Conversely, according to the Euler–Bernoulli theory, Bonopera and Chang (2021) exploited static small-deflection measurements for predicting residual prestressing in PC girder-bridges instead of the use of natural frequencies. Indeed, both frequencies and mode shapes are not significantly affected by the prestressing changes even with none-to-slight concrete damage (De Angelis et al. 2024; Gandelli et al. 2024). Yet, Bonopera and De Matteis (2026) revised the above mentioned “static deflected shape” method according to the Timoshenko theory. Numerical methods were also utilized to study the behavior of PC girder-bridges by focusing on distinct aspects. According to experiments on PC girder-specimens, Finite-Element (FE) analyses showed to be able to identify both their structural and local response (i.e., end anchorages, tendon release, tensile cracking) with a proper accuracy (Yapar et al. 2015; Van Meirvenne et al. 2018; Lee et al. 2020). FE simulations were also implemented to consider the long term effects (concrete creep, curing, shrinkage, and tendon relaxation) on the behavior of PC girder-bridges with bonded (Lou et al. 2014; Bonopera et al. 2022) or unbonded tendons (Lou et al. 2013). Gan et al. (2019) used FE analyses to sustain that the divergence declared by Noble et al. (2015) and (2016), that the dynamic effect of a compressive force and that of a prestressing on a concrete beam are phenomenologically different, was caused by the closure of shrinkage cracks and/or microcracks. Considering such an unclear correlation between frequency and prestressing force, the conclusions furnished by Noble et al. (2015) were revised by Bonopera et al. (2023). These scholars made additional FE analyses on post–tensioned steel girders to prevent the stiffening effects due to the microcrack closure and time-increment of concrete elastic modulus (Jaiswal 2008; Bonopera et al. 2019 and 2021). They determined that the prestressed beam dynamics depends on the shear deformation and tendon contact with the surrounding beam section. If the cables are not in contact with the cross-section, the beam dynamics due to a compressive force is coincident to that caused by a low value of prestressing. Therefore, the prestressed beam dynamics is initially ruled by the compression-softening effect. Numerical models of PC girder-bridges with bonded and/or unbonded tendons were additionally proposed. E.g., during a set of laboratory experiments, Consiglio et al. (2026) simulated the nonlinear statics of a full-scale pre– tensioned concrete beam with straight tendons using a Two-Dimensional (2D) FE modeling including shear deformation. Other researchers instead compared the response of PC girder-bridges, with bonded and internally unbonded tendons, obtaining important differences in terms of cracking as well as ultimate load capacity (Pang et al. 2022). Specifically, a FE analysis was made to study the influence of prestressing levels on the flexural behavior of PC girders with unbonded tendons, revealing that higher prestressing results in higher ultimate load-carrying capacity even if with lower deflections (Le et al. 2020). However, to the authors’ best knowledge, an additional numerical work on PC girder-bridges considering shear deformation is necessary to clarify their statics in presence of second-order effects. In fact, most of the aforementioned studies were developed based on the Euler–Bernoulli theory and/or 2D FE modeling, i.e., in detriment of the Timoshenko theory and/or Three-Dimensional (3D) FE modeling.