PSI - Issue 84

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

460

specimen, whilst their elastic modulus was assumed equal to 200 GPa. Poisson’s ratio υ = 0.25. Carriage -hinge external constraints, to have the pinned-end conditions, were assigned on a row of nodes at L = 6.62 m [Fig. 2 in Bonopera et al. (2018)]. To obtain the second-order small-deflections [ v tot,shear,3D FE ( a ) ( x )], at i = 1, …., 7, a two-step FE procedure was performed. The post–tensioning force ( N 0 ) was eccentrically applied at both specimen ends as an external normal face pressure and, in particular, on the anchor blocks. Therefore, second-order initial curvature and axial deformation of the girder–tendon system were achieved. In short, N 0 was treated as a compressive force, as takes place in the magnification factor formula of the second-order shear effects (Section 4). Consequently, at each post–tensioning ( N 0 ), a vertical load ( F ) was likewise applied at the midspan as a normal face pressure, and with a magnitude of ≈ 20.0 kN, then gradually incremented t o ≈ 22.5 and ≈ 25.0 kN . Subsequently, the second-order deflections [ v tot,shear,3D FE ( a ) ( x )] were gained for the corresponding new deformed configurations and with respect to the second-order initial cambers [Fig. 1(a)]. Besides, to gain the numerical first-order critical buckling load, N crE,shear,1,3D FE , for each test execution day, buckling analyses were executed [Fig. 1(b)]. Similarly to the post–tensioning, the compressive force was assigned as an external face pressure on both anchor blocks [Fig. 1(b)]. In this case, all the self-unit weights of the girder–tendon system were not accounted for. Thus, a n average percentage error Δ = ( N crE,shear,1,3D FE − N crE,shear,1 ) / N crE,shear,1 equal to − 9.6% was obtained (Table 1). N crE,shear,1 is the reference PC girder- bridge specimen’s first-order critical buckling load [Eq. (4) in Bonopera and De Matteis (2026)]. Artificial geometric imperfections were not included for both nonlinear static and buckling analyses in Strand7 (2010). Notably, the cross-sectional area of the 7 strands ( A tendon ) was assumed as unchanged after deformations. The average measurements of time-dependent elastic moduli ( E aver and G aver ), post– tensioning forces ( N 0 and N x ), vertical load ( F ) and small-deflection measurements ( v i ), recorded during the three-point bending experiments, were listed in Table 2.

(a)

(b)

Fig. 1. Solid FE modeling in Strand7 (Section 3). PC girder-bridge specimen at the concrete age of 433 days ( E aver = 38,791 MPa): (a) Nonlinear deflected shape for N x = 820 kN and F = 25.1 kN; (b) First-order critical buckling shape ( N crE,shear,1,3D FE = 10,531 kN).

Table 1. Measured average time-dependent elastic moduli ( E aver and G aver ) and first-order critical buckling loads ( N crE,shear,1 and N crE,shear,1,3D FE ) for each test execution day. Age of concrete (days) Age of post– tensioning (days) N 0,aver (kN) E aver (MPa) G aver (MPa) Var. (%) N crE,shear,1 (kN) N crE,shear,1,3D FE (kN) Var. (%)

28

–

– 30,560 12,733 –

–

–

–

426

1

618 34,870 14,529 +14.1 10,471

9,479

–9.5

427

2

722 37,618 15,674 +23.1 11,296

10,216

–9.6

433

8

820 38,791 16,163 +26.9 11,648

10,531

–9.6

4. Reference solution including shear deformation (Bonopera and De Matteis 2026) The reference solution of the “static deflected shape” method, which focuses on a simply supported PC girder bridge of length L , is depicted in Fig. 1(a–c) in Bonopera et al. (2018). The prismatic beam is externally subjected to an eccentric prestressing force ( N ) and afterwards to a vertical load ( F ) at its midspan. Concrete chord elastic modulus