PSI - Issue 78

Marco Bonopera et al. / Procedia Structural Integrity 78 (2026) 1143–1150

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Gan et al. 2019). Structural engineers should be able to estimate such changes in prestressed structures over the course of their design life to ensure their safety. E.g., vibration measurements are useful applications to control their calculation model, evaluate their stiffness or monitor their stresses (Yang et al. 2018). Particularly, Hamed and Frostig (2006), Jaiswal (2008), Limongelli et al. (2016) and Bonopera et al. (2019) declared that the stiffness of Prestressed Concrete (PC) girder-bridges, with a parabolic or a straight tendon, only significantly change under the effect of crack initiation or re-opening. Previously, Noble et al. (2015 and 2016) performed several experiments on a number of post – tensioned steel and concrete girders in small-scale. The researchers found that the small reduction in fundamental frequency with increasing prestressing force is not related to the softening effect. Therefore, they deduced that the compression-softening theory must be eliminated from discussion of all forms of prestressed members. Moreover, the researchers claimed that the dynamic effect of a compressive force and that of a prestressing on a beam are different on a phenomenological level. Gan et al. (2019) sustained that such a divergence was caused by the closure influence of the shrinkage cracks and/or microcracks inside the PC girders that, in turn, was not assumed within the analytical predictions of the fundamental frequency. Considering the unclear correlation between the fundamental frequency and prestressing force, the conclusions provided by Noble et al. (2015) were revised by Bonopera et al. (2023). These scholars executed additional investigations on simply supported post – tensioned steel girders with the goal to prevent the stiffening effects caused by the microcrack closure and time-increment of the concrete elastic modulus (Jaiswal 2008; Bonopera et al. 2019 and 2021). Consequently, they found that the prestressed beam dynamics strongly depends on the contact of the tendons with the surrounding beam’s section. If the cables are not in contact with the surrounding cross-section, the beam dynamics due to a compressive force is coincident to that caused by a low value of prestressing force. Thus, the dynamics of prestressed members are initially ruled by the compression-softening effect. The aforementioned works were all based on the Euler – Bernoulli theory, whilst studies employing the Timoshenko beam theory were much fewer. In fact, shear deformation is usually neglected in applications of structural engineering. Hence, this work was necessary for better clarifying the behavior of PC girder-bridges in presence of second-order shear effects. In this work, a reference model comprising a simply supported Timoshenko beam (Bažant and Cedolin 2010) and prestressed by an eccentric straight tendon was implemented. The post – tensioning force was assumed to be an external axial load applied eccentrically to the beam ends. Subsequently, the deflection measured from a three-point bending test, conducted on the aforementioned PC girder-bridge specimen, was approximated by multiplying the first-order deflection by the magnification factor of the second-order shear effects (Timoshenko and Gere 1961; Bažant and Cedolin 2010) based on the compression-softening theory. First, small-deflection measurements along the specimen, gained from 27 three-point bending tests, and subjected to different values of post – tensioning, were examined to verify the beam model ’s accuracy. Finite-Element (FE) simulations assuming shear deformation and geometric nonlinearities were also implemented to analyze such an issue. Consequently, the nondestructive method proposed by Bonopera and Chang (2021) was revised based on the Timoshenko beam theory and employed to identify the existing post – tensioning forces. According to the obtained findings, the shear deformation should strongly be assumed for a better accuracy in most simulations of structural modeling, and estimating the prestressing force using small-deflection measurements. 2. Reference solution including shear deformation Figure 1 in Bonopera et al. (2018) illustrates the formulation of the reference solution which focuses on a simply supported prismatic PC girder-bridge of length L . The member is first subjected to an eccentric post – tensioning force N (with eccentricity e ) with respect to the centroid of its cross-section [Fig. 1(a) in Bonopera et al. 2018] and subsequently to a vertical load F at its midspan [Fig. 1(b) in Bonopera et al. 2018]. The post – tensioning ( N ) is assumed to be externally applied. The chord elastic modulus of concrete E , cross-sectional area A and cross-sectional second moment of the area I are known parameters. The initial deflected-shape curve v (0) after the application of the eccentric post – tensioning force ( N ) can be expressed by Eq. (1) in Bonopera et al. (2018). Subsequently, a vertical load ( F ) is applied at the midspan. Thus, the corresponding bending moments at the left and right portions of the girder are respectively expressed by Eqs. (2a) and (2b) in Bonopera et al. (2018). Incorporating such equations into the formula of the beam axis curvature M = – EI d 2 v (1) /d x 2 furnishes the expression v (1) = v (0) + v tot ( a ) , where v tot ( a ) is the deflected shape curve of the Euler – Bernoulli beam under the post – tensioning N and vertical load F [Fig. 1(c) in Bonopera et al.