PSI - Issue 84

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

470

Matteis (2026), as illustrated in Section 4. Such second-order deflections [ v tot,shear ( a ) ( x )] were denoted as “ Analytical – Shear ”. Notably, the long-term post–tensioning losses were –14.0, –14.0 and –14.2% respectively (Table 2). To obtain the deflections v tot,shear ( a ) ( x ), the solid FE modeling was utilized to determine the parameters of the magnification factor formula for each test day (Sections 3–4). The first-order critical buckling load, N crE,shear,1,3D FE = 9,806 kN [Fig. 1(b)]. Instead, the first-order deflections [ v I,shear,3D FE ( a ) ( x )], at i = 1, …., 7, were gained for each test combination, as illustrated in Section 3 [Fig. 1(a)]. In Table 2, they were denoted as “ Solid FE model (I-order) ”. Within the magnification factor formula, the applied post–tensioning N x 1 were respectively accounted for magnifying the first-order deflections v I,shear,3D FE ( a ) ( x ) at i = 1, 2 and 3. Conversely, the post–tensioning N x ,aver were used for magnifying the midspan deflections v I,shear,3D FE ( a ) ( x ), i.e., at i = 4, whilst post–tensioning N x 2 were utilized for magnifying the deflections v I,shear,3D FE ( a ) ( x ) at i = 5, 6 and 7 (Table 2). Thus, comparing all displacements, the absolute average error between second-order small-deflections [ v tot,shear ( a ) ( x )] and deflection measurements ( v i ) was 0.28 mm, corresponding to an average percentage error of 9.5%. Vice versa, considering the cross-sections at i = 2 and 4, a maximum percentage error of 9.2% was obtained among the 11 test combinations (Table 2). Notably, as well in the first article of this work, the non-scattered errors from the above comparisons were caused by the poor mesh size resolutions used within the FE modeling of the PC girder-bridge [Fig. 1(a)]. In fact, when the magnification factor formula was used for magnifying a set of analytical first-order displacements v I,shear ( a ) ( x ) [Eq. (1) in Bonopera and De Matteis (2026)], the absolute average error between deflections v tot,shear ( a ) ( x ) and v i was 0.04 mm (corresponding to an average error of −1.5% ). Accordingly, since the maximum post–tensioning ( N x ,aver,max =515 kN; Table 2) was 4.7% of the reference buckling load, N crE,shear,1 I , Nx =0 =10,869 kN [Eq. (1)], the compression-softening theory assuming shear deformation was confirmed for PC girder-bridges under long-age conditions when an important cracking is precluded. More detailed information on the above measurements ( v i ) were described in Bonopera and Chang (2021). 6. Effective time-dependent post–tensioning identification according to the “static deflected shape” method The effective time-dependent post–tensioning, N x 1 and N x ,aver , at 288, 290 and 291 days were identified for each test combination (Table 2) according to the “static deflected shape” method (Section 4). Equation (1), reported in the first article, was employed. N x 1 was the post–tensioning at the PC girder-bridge’s left support [Fig. 1 in Bonopera and Chang (2021)]. Tables 3–4 show the post–tensioning forces, N a 1,shear,3D FE and N a ,aver,shear,3D FE , evaluated by the quarter ( v 2 ) and midspan deflection measurements ( v 4 ) respectively (Tests 1 and 3). The measurements ( v 2 and v 4 ) were separately counted for the second-order small-deflections v tot,shear ( x ) ( x ). N crE,shear,1,3D FE =9,806 kN [Fig. 1(b)]. Moreover, the quarter first-order small-deflections, v I,2,shear,3D FE , were those listed in Table 2 [ Solid FE model (I-order) ]. Yet, the midspan first-order ones, v I,4,shear,3D FE , were obtained by the same solid FE modeling (Section 3) in which the PC girder-bridge (and its cross-section ends) were conversely modeled using 351,808 brick elements and with a mesh size = 12.25 × 12.4 × 15 mm 3 . Therefore, the comparisons between applied and identified post–tensioning were expressed by the percentage errors Δ = ( N a 1,shear,3D FE − N x 1 ) / N x 1 (Table 3) and Δ = ( N a ,aver,shear,3D FE − N x ,aver ) / N x ,aver (Table 4). Additionally, Table 3 compares the identified post–tensioning, N a 1,shear,3D FE , with the corresponding ones, N a 1 , estimated according to the method when based on the reference Euler–Bernoulli solution (Test 2). Similarly, Table 4 compares the post–tensioning, N a ,aver,shear,3D FE , with the related ones, N a ,aver (Test 4) [Tables 5–6 in Bonopera and Chang (2021)]. As a result, N a ,aver,shear,3D FE were satisfactorily identified because the midspan deflections, v I,4,shear,3D FE , were numerically computed by a higher mesh size resolution, i.e., that equal to 12.25×12.4×15 mm 3 . Notably, in terms of errors (Δ >15%), the identifications N a ,aver,shear,3D FE worsened corresponding to the applied vertical loads F ≥ 42.2 kN (Test 3). With this regards, a mesh sensitivity analysis for solid FE modeling of PC girder-bridges will be developed with the aim to improve the identification accurancies (Galano et al. 2023). More detailed information were illustrated in Bonopera and Chang (2021). Table 3. Comparison between post–tensioning force identifications ( N a 1,shear,3D FE and N a 1 ) and measured and estimated parameters for each test combination obtained using quarter deflection measurements ( v 2 ) (Tests 1–2). Test 1 Test 2 deflection at a quarter ( v 2 ) Solid FE modeling “Euler–Bernoulli” beam model