PSI - Issue 84

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

462

Table 2. Comparison between second-order small-deflections [ v tot,shear,3D FE ( a ) ( x )] and small-deflection measurements ( v i ) for each test execution day and related to the layout depicted in Fig. 5 in Bonopera et al. (2018). ( a ) ( x ) and v tot,shear

Age of concrete (days)

E aver

G aver

N 0

N x

F

v 1

v 2

v 3

v 4

v 5

v 6

v 7

(MPa)

(MPa)

(kN) (kN) (kN)

(mm) (mm) (mm) (mm) (mm) (mm) (mm)

Analytical – Shear 1.03 1.94 2.58 2.83 2.58 1.94 1.03 Solid FE model 0.96 1.81 2.40 2.63 2.40 1.81 0.96 LVDT 1.45 1.95 2.62 2.84 – 1.93 1.02 Analytical – Shear 1.16 2.17 2.89 3.16 2.89 2.17 1.16 Solid FE model 1.11 2.08 2.76 3.03 2.76 2.08 1.11 LVDT 1.59 2.20 2.95 3.20 – 2.17 1.15 Analytical – Shear 1.28 2.40 3.19 3.50 3.19 2.40 1.28 Solid FE model 1.20 2.26 3.01 3.29 3.01 2.26 1.20 LVDT 1.42 2.32 3.12 3.43 3.03 2.27 1.22 Analytical – Shear 0.96 1.80 2.39 2.62 2.39 1.80 0.96 Solid FE model 0.90 1.68 2.24 2.45 2.24 1.68 0.90 LVDT 1.19 1.78 2.39 2.59 2.33 1.73 0.94 Analytical – Shear 1.08 2.02 2.69 2.95 2.69 2.02 1.08 Solid FE model 1.04 1.94 2.59 2.84 2.59 1.94 1.04 LVDT 1.31 2.00 2.67 2.92 2.60 1.94 1.05 Analytical – Shear 1.20 2.24 2.99 3.27 2.99 2.24 1.20 Solid FE model 1.13 2.11 2.82 3.08 2.82 2.11 1.13 LVDT 1.46 2.22 2.97 3.23 2.90 2.16 1.17 Analytical – Shear 0.94 1.76 2.35 2.57 2.35 1.76 0.94 Solid FE model 0.88 1.65 2.19 2.40 2.19 1.65 0.88 LVDT 1.20 1.75 2.33 2.54 2.29 1.71 0.92 Analytical – Shear 1.07 2.00 2.66 2.92 2.66 2.00 1.07 Solid FE model 1.02 1.90 2.52 2.76 2.52 1.90 1.02 LVDT 1.33 1.98 2.65 2.88 2.60 1.94 1.04 Analytical – Shear 1.17 2.19 2.92 3.20 2.92 2.19 1.17 Solid FE model 1.10 2.06 2.74 3.00 2.74 2.06 1.10 LVDT 1.42 2.17 2.91 3.17 2.86 2.14 1.15

619 620 20.2

426

34,870 14,529

619 620 22.6

615 617 25.0

723 724 20.1

427

37,618 15,674

720 721 22.6

720 721 25.1

820 820 20.2

433

38,791 16,163

820 820 22.9

820 820 25.1

6. Effective post–tensioning identification according to the “static deflected shape” method With the aim to identify the effective post–tensioning ( N x ) at 426, 427 and 433 days of concrete age (Table 2), the solid FE modeling (Section 3) was utilized to compute the parameters of the magnification factor formula of the second-order shear effects in Eq. (1), according to the “static deflected shape” method (Section 4). Specifically, Table 3 shows the post–tensioning forces ( N a ,shear,3D FE ) estimated by the quarter small-deflection measurements ( v 2 ) for each test combination (Test 1). In details, the deflections v 2 were counted as parameters for the second-order small deflections v tot,shear ( x ) ( x ) in Eq. (1). N crE,shear,1,3D FE was instead the numerical first-order critical buckling load of the PC girder-bridge specimen, which was obtained by the buckling analyses [Fig. 1(b)] and likewise determined for each test execution day (Table 1). Furthermore, the first-order small-deflections [ v I,shear,3D FE ( a ) ( x )], at i = 2, were calculated using the same PC girder-bridge’s solid FE modeling. Particularly, the related vertical loads ( F ) were applied at its midspan as a normal face pressure and with their corresponding magnitudes (Table 2). Subsequently, the quarter first order small-deflections [ v I,2,shear,3D FE ] were computed regarding the linear deflected shapes and with respect to the undeformed configurations of the girder–tendon system. All self-unit weights were not accounted for in Strand7 (2010). The comparisons between the applied ( N x ) and the identified post–tensioning ( N a ,shear,3D FE ) were displayed by the percentage errors Δ = ( N a ,shear,3D FE − N x ) / N x . Besides, Table 3 compares the identified post–tensioning forces ( N a ,shear,3D FE ) with the corresponding ones ( N a ,shear ) estimated when the parameters of the magnification factor formula were computed by the reference solution including shear deformation [Table 3 in Bonopera and De Matteis (2026)]. This analytical configuration was underlined as Test 2. Accordingly, the poor predictions of the effective post– tensioning ( N a ,shear,3D FE ) were mainly caused by the insufficient mesh size resolutions used within the solid FE modeling of the PC girder-bridge [Figs. 1(a–b)], which prevented an accurate computation of the first-order critical