PSI - Issue 84

Marco Bonopera et al. / Procedia Structural Integrity 84 (2026) 465–472

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and static deformations with changing prestressing in bridges is a crucial issue (Gan et al. 2019). Structural engineers should be able to evaluate such variations to guarantee their load-carrying capacity. Tests are the most useful way to investigate both their flexural and shear response. Some works studied the flexural response of full-scale Prestressed Concrete (PC) girder-bridges (Wang et al. 2021; Kralovanec et al. 2022), while others focused on their shear one (Osborn et al. 2012; Lantsoght et al. 2021). These studies demonstrated how PC girder-bridges with none-to-slight damage ensured an adequate load-bearing capacity. Nonetheless, experiments on full-scale PC girder-beams could strongly be challenging (Bagge et al. 2018). This being the reason why reduced-scale PC girder-bridges were commonly investigated (Deng et al. 2001). For instance, vibration measurements were proper ways to evaluate their stiffness. Jaiswal (2008) and Bonopera et al. (2019) established that the PC girder-bridges’ stiffness only changes under the influence of crack initiation or re-opening. Noble et al. (2016) carried on tests on a series of reduced-scale PC girder-bridges. These scholars proved that the mild decrement in fundamental frequency with the increment in prestressing is not related to the softening effect. As a result, the researchers deduced that the compression-softening theory must be removed from discussion of all forms of prestressed beams. Vice versa, Bonopera and Chang (2021) utilized small-deflection measurements for identifying prestressing based on the nonlinear statics of the Euler– Bernoulli beam. Natural frequencies are indeed unaffected by the variations in prestressing even with none-to-slight concrete damage (De Angelis et al. 2024). Numerical approaches were also used to investigate the structural response of PC girder-bridges. Based on a set of experiments, Finite-Element (FE) simulations proved to assess both their global and local response (Lee et al. 2020). FE analyses were developed to take the long-term effects into account on the flexural behavior of PC girders with bonded or unbonded tendons (Lou et al. 2013; Bonopera et al. 2022). Gan et al. (2019) adopted FE analyses to claim that some divergences on the PC beam dynamic, sustained by Noble et al. (2015) and (2016), were due to the closure of shrinkage cracks and/or microcracks. Therefore, the conclusions provided by Noble et al. (2015) were additionally investigated by Bonopera et al. (2023). These researchers made a set of FE analyses on post–tensioned steel girder bridges to avoid the stiffening effects caused by the microcrack closure and time-increment concrete elastic modulus (Jaiswal 2008; Bonopera et al. 2019). They obtained that the prestressed beam dynamics is related to the shear deformations and tendon contact with the surrounding beam section. If the cables do not touch the cross-section, the beam dynamics caused by a compressive force coincides to that due to a low prestressing. Accordingly, the initial dynamics of the prestressed beam is dominated by the compression-softening effect. Numerical models were also presented. Consiglio et al. (2026) analyzed the nonlinear statics of a pre–tensioned concrete beam with straight bonded tendons through a Two-Dimensional (2D) FE modeling which included shear deformation. Other scholars compared the global structural response of PC girders by achieving significant differences in terms of cracking as well as ultimate load capacity (Coronelli et al. 2023). Particularly, a FE analysis investigated the prestressing effect on the flexural response of PC girders with unbonded tendons. It was revealed that higher prestressing results in higher ultimate load-bearing capacity even if with lower deflections (Le et al. 2020). Huang et al. (2018) instead proposed a Three-Dimensional (3D) FE model representing the long-term behavior of PC girders for improving the numerical modeling accuracy against experimental data. However, this work on PC girders, composed of two articles, and focusing on shear deformation, is needed to better explain their statics in presence of second-order effects. Indeed, most of the above works were implemented according to the Euler–Bernoulli theory and/or 2D FE modeling, i.e., instead of the assumption of the Timoshenko theory and/or 3D FE modeling. In this second article, an additional high-fidelity solid FE model including a concrete girder-bridge specimen, and post–tensioned by a parabolic bonded tendon, was similarly developed. Geometric nonlinearities and time-dependent post–tensioning losses were considered. The simply supported specimen was mainly constructed using solid FEs in Strand7 (2010). Subsequently, comparisons with three-point bending executed on the above reduced-scale PC girder bridge (Bonopera et al. 2021), distinguished by a significant slenderness ratio, high-strength concrete, and subjected to different levels of post–tensioning, were carried out. Then, with the goal to estimate the effective post–tensioning, the FE model was directly used to calculate the parameters of the magnification factor formula of the second-order shear effects. According to the “static deflected shape” method, described in the first article, the shear deformation should firmly be considered for prestressing identifications. This solution strongly needs assuming high-mesh size