PSI - Issue 84

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

459

In this article, a high-fidelity solid FE model including a concrete girder-bridge specimen, and post–tensioned by an eccentric straight tendon, was implemented to study the second-order shear deformation in PC girder-bridges. Geometric nonlinearities were assumed, while the simply supported concrete girder-specimen was mostly elaborated through solid FEs. Particularly, the post–tensioning force was applied as an external load assigned to the beam ends. The Strand7 program (2010) was adopted. Subsequently, comparisons with three-point bending tests conducted on the aforementioned reduced-scale PC girder-bridge (Bonopera et al. 2018), characterized by a significant slenderness ratio, high-strength concrete, and subjected to different initial post–tensioning levels, were elaborated. Afterwards, with the goal to evaluate the post–tensioning, the FE model was employed to determine the parameters of the magnification factor formula of the second-order shear effects, according to the “static deflected shape” method (Bonopera and De Matteis 2026). Here the pros and cons of the use of different FE modeling, reported in literature, were commented. As a result, the shear deformation in PC girder-bridges should highly be assumed for gaining better accuracies, in terms of results, in most evaluation techniques. Moreover, in the second article of this work, experimental outcomes regarding a concrete girder-bridge specimen with parabolic tendon (Bonopera et al. 2021), and subjected to time dependent post–tensioning losses, were similarly compared with a solid FE modeling. 2. Concrete girder-bridge specimen post–tensioned by a straight tendon The simply supported PC girder-bridge specimen utilized by Bonopera et al. (2018), composed of a straight tendon, and with a high-strength concrete’s unit weight of 22.71 kN/m 3 , was considered [Fig. 3 in Bonopera et al. (2018)]. Its slenderness ratio was 57, whilst the length/height ( L /h) ratio was 17. The specimen ends had a rectangular cross section (450 × 600 mm 2 ) and a length of 160 mm at the supports [Fig. 2 in Bonopera et al. (2018)]. Their cross-section ends were closed with a 450 × 600 × 40 mm 3 steel plate. The latter allowed the passage and the anchorage of the 7 straight strands (with a steel’s unit weight = 76.65 kN/m 3 ) by a 150 × 150 × 60 mm 3 steel anchor block of 70 N in weight. Besides, a 150 × 150 × 120 mm 3 load cell, of 78.5 N in weight, was located between the rectangular plate and the block at one of the specimen ends to measure the applied post–tensioning forces, N 0 and N x . The three-point bending tests required the positioning of nine Linear Variable Differential Transformer (LVDT) sensors at specific cross-sections along the specimen, , i.e., at i = 0, …, 8 [Fig. 5 in Bonopera et al. (2018)]. Moreover, the time-dependent chord elastic ( E aver ) and shear modulus ( G aver ) were estimated by compression tests at 28 days and at each execution day of the three-point bending, i.e., at 426, 427 and 433 days of the concrete age [Table 2 in Bonopera and De Matteis (2026)]. More detailed information on the experimental set-up, instrumentation, measurements, procedures and elastic moduli evaluation were described in Bonopera et al. (2018) and Bonopera and De Matteis (2026). 3. Solid finite-element modeling Solid FE modeling was conducted to compare the small-deflection measurements, v i , at i = 1, …., 7 [Fig. 5 in Bonopera et al. (2018)]. The spatial solution of the PC girder-bridge specimen was discretized into a group of FE linked to each other in a specific manner. In fact, FE analyses can solve problems of PC or steel girder-bridges involving complicated geometries and/or boundary conditions (Bonopera et al. 2023). Linear elastic material models, geometric properties and self-unit weights (Section 2) were accounted for in Strand7 (2010). Transverse shear deformations were assumed, whilst rotary inertia was neglected. Particularly, the specimen and its cross-section ends were modeled using 58,644 brick elements with a mesh size = 25 × 25 × 20 mm 3 . The 7 strands were instead modeled with a diameter of 13.3 mm and discretized into 355 truss elements for a length of 7,100 mm per each. The latter passed internally along the specimen according to its design. Specifically, they had common nodes with the specimen body since they were always in contact with the surrounding cross-section. Here, the small eccentricity of the tendon ( e / h = 0.125) was the distance of the central strand from the girder centerline. Furthermore, the nine 180 × 300 × 7.5 mm 3 steel plates used to locate each LVDT sensor were modeled using 972 plate/shell elements with a mesh size = 25 × 20 mm 2 [Fig. 4(b) in Bonopera et al. (2018)]. Each plate was 31.3 N in weight. Conversely, the 80 × 150 × 75 mm 3 upper part of the steel supports were modeled using 144 brick elements with a mesh size = 20 × 25 × 25 mm 3 [Fig. 2 in Bonopera et al. (2018)]. Rectangular plates (1,728 elements), anchor blocks (216 elements) and load cell (216 elements) were likewise modeled using brick elements (with a mesh size = 25 × 25 × 20 mm 3 ). The meshes between strands, LVDT’s plates, supports, rectangular plates, anchor blocks and load cell were rigidly connected along the