PSI - Issue 84

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

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resolutions within the solid FE model of the girder–tendon system. Conclusions and future developments were also reported in the last section. 2. Concrete girder-bridge specimen post–tensioned by a parabolic bonded tendon The simply supported PC girder-bridge used by Bonopera et al. (2021), composed of a parabolic bonded tendon, was taken into consideration [Fig. 1 in Bonopera et al. (2021)]. Its slenderness ratio was 60, while the length/height ( L /h) ratio was 17. The specimen ends had a rectangular cross-section with an area of 441 × 595.2 mm 2 and a length of 300 mm at the supports [Fig. 2(a) in Bonopera and Chang (2021)]. Such cross-section ends respectively permitted the anchorage of the 7 parabolic strands together with a 147 × 148.8 × 60 mm 3 steel anchor block of 70 N in weight. Furthermore, a 147 × 148.8 × 120 mm 3 load cell, of 78.5 N in weight, was respectively settled between the concrete ends and the blocks to measure the post–tensioning forces, N 0 x 1 , N 0 x 2 , N x 1 and N x2 . The three-point bending tests, conducted after the occurrence of the long-term post–tensioning losses, required the fastening of ten displacement transducers at specific positions along the specimen, , i.e., at i = 0, …, 8 [Fig. 1 and Fig. 2(b) in Bonopera and Chang (2021)]. Besides, the time-dependent initial tangent elastic modulus ( E cm,aver ) was estimated by compression tests at 28 days and, additionally, on a number of cores drilled at the quarter and midspan cross-sections (at i = 2, 4 and 6) after three-point bending. I.e., at 431 days of concrete age [Fig. 2(c) and Table 2 in Bonopera and Chang (2021)]. Vice versa, the determination of the time-dependent initial tangent shear modulus, G cm,aver = E cm,aver /[2(1+ υ)], was reported in Table 1. Poisson’s ratio υ = 0.2. More detailed information on the above experimental layout, instrumentation and elastic modulus evaluation were illustrated in Bonopera et al. (2021) and Bonopera and Chang (2021). 3. Solid finite-element modeling Similarly to the first article of this work, solid FE modeling was executed in Strand7 (2010). Indeed, the small deflections v i , at i = 1, …., 7 [Fig. 1 in Bonopera and Chang (2021)], measured after the occurrence of the time dependent post–tensioning losses (Section 2), were compared. Linear elastic material models and geometric properties were accounted for. Rotary inertia was neglected, while transverse shear deformations were assumed (Chen and Leon 2019). The specimen (and its cross-section ends) were modeled using 43,976 brick elements with a mesh size = 24.5 × 24.8 × 30 mm 3 . Conversely, the bonded tendon was modeled by an unique cable with a dimension of 35.2 mm in diameter ( A tot,tendon = 973 mm 2 ) and discretized into 239 truss elements (mesh size = 30 mm) for an effective parabolic length L tendon = 6.886 m. The FE cable internally had common nodes with the specimen body since the 7 strands were always in contact with the surrounding cross-section. Here, the small eccentricities of the bonded tendon at the simply supports ( e 1 /h=−0.25) and midspan ( e 2 /h=0.25) represented the distance of the parabolic cable from the cross-section centroid [Fig. 3 in Bonopera and Chang (2021)]. Moreover, the ten 210 × 294 × 7.5 mm 3 steel horizontal plates, utilized to arrange each displacement transducer, were modeled using 840 plate/shell elements with a mesh size = 30 × 24.5 mm 2 [Fig. 2(b) in Bonopera and Chang (2021)]. Each plate was 31.3 N in weight. Contrariwise, the 120 × 147 × 74.4 mm 3 upper part of the steel supports were modeled using 144 brick elements with a mesh size = 30 × 24.5 × 24.8 mm 3 [Fig. 6 in Bonopera et al. (2021)]. Anchor blocks (144 elements) and load cells (288 elements) were similarly modeled adopting brick elements and with a mesh size = 24.5 × 24.8 × 30 mm 3 . Summarizing, the meshes between parabolic cable, horizontal plates, supports, anchor blocks and load cells were rigidly connected. Their elastic modulus E s = 200 GPa. Poisson’s ratio υ = 0.25. The modeling of the servo velocity seismometers, of 270 g in lightweight, was instead neglected. Carriage-hinge external restraints, to have the simple supports, were applied on a row of nodes at L = 6.87 m. Firstly, the first-order small-deflections [ v I,shear,3D FE ( a ) ( x )], at i = 1, …., 7, were obtained. The corresponding vertical loads ( F ) were assigned at the midspan as an external normal face pressure and with their corresponding magnitudes (Table 2). The influence area of the vertical loads ( F ) was A F = 245 × 330 mm 2 . Specifically, the first magnitude F 1 ≈ 24.5 kN, then gradually incremented to F 2 ≈ 34 .0, F 3 ≈ 42 .5 and F 4 ≈ 51 .5 kN (Table 2). Subsequently, the first-order deflections [ v I,shear,3D FE ( a ) ( x )] were extrapolated for the new linear deflected shapes (and axial deformations) and with respect to the undeformed configurations of the girder–tendon system [Fig. 1(a)]. Secondly, to determine the numerical first-order critical buckling load, N crE,shear,1,3D FE = 9,806 kN, for each test day of three-point bending, i.e., at 421, 423