PSI - Issue 84

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

468

and 424 days of concrete age, a buckling analysis was performed [Fig. 1(b)]. The post–tensioning force ( N 0 x ) was similarly assigned at both concrete ends (as a compressive force) using a normal face pressure and, specifically, applied on the anchor blocks [Fig. 1(b)]. Shortly, the configuration which takes place in the magnification factor formula was assumed (Section 4). Thus, a percentage error Δ =( N crE,shear,1,3D FE − N crE,shear,1 I , Nx =0 ) / N crE,shear,1 I , Nx =0 = − 9.8% was gained (Table 1). N crE,shear,1 I , Nx =0 was the reference PC girder-bridge specimen’s critical buckling load [Eq. (1)]. Artificial geometric imperfections were not induced in both linear static and buckling analyses (Fig. 1) (Ilanko 1990). Notably, the cross-sectional area of the 7 parabolic strands ( A tot,tendon ) was considered unchanged after deformations. Moreover, the average measurements of time-dependent elastic moduli ( E cm,aver = 36,054 MPa and G cm,aver = 15,023 MPa) were itemized in Table 1. Vice versa, the post–tensioning forces ( N 0 x 1 , N 0 x 2 , N x 1 and N x 2 ), vertical loads ( F ) and small deflections v i , at i = 1, …., 7, measured during three-point bending, were listed in Table 2.

(a)

(b)

Fig. 1. Solid FE modeling in Strand7. PC girder-bridge specimen with a parabolic bonded tendon at the concrete age of 424 days ( E cm,aver = 36,054 MPa): (a) Linear deflected shape for F = 51.9 kN; (b) First-order critical buckling shape ( N crE,shear,1,3D FE = 9,806 kN).

Table 1. Average time-dependent initial tangent elastic moduli ( E cm,aver and G cm,aver ) and first-order critical buckling loads ( N crE,shear,1 I , Nx =0 and N crE,shear,1,3D FE ) for each test day. Age of concrete (days) Age of post– tensioning (days) N 0 x ,aver (kN) E cm,aver (MPa) G cm,aver (MPa) Var. (%) N crE,shear,1 I , Nx =0 (kN) N crE,shear,1,3D FE (kN) Var. (%)

28

–

–

35,198 14,666 –

–

–

–

421 423 424

288 290 291

516 516 515

36,054 15,023 +2.4

10,869

9,806

–9.8

4. Reference solution including shear deformation (Bonopera and De Matteis 2026) The model presented in Fig. 7 in Bonopera et al. (2021), i.e., a simply supported Euler–Bernoulli beam of length L = 6.87 m, is the reference solution of the “static deflected shape” method. The concrete beam with a rectangular cross section ( A ) is externally subjected to the eccentric prestressing force N 0 x ,aver ( e 1 : tendon’s small eccentricity) and distributed load q 0 x ,aver due to the parabolic tendon under prestressing ( N 0 x ,aver ). Its time-dependent initial tangent elastic modulus ( E cm,aver ) and geometric properties are known values. v (0) is the initial nonlinear camber which occurs after the application of prestressing ( N 0 x ,aver ) and distributed load ( q 0 x ,aver ). Subsequently, a vertical load ( F ) is assigned at the midspan of the initial deflected shape v (0) [Fig. 11 in Bonopera et al. (2021)]. Thus, v (1) is the nonlinear deflected shape under prestressing N 0 x ,aver , distributed load q 0 x ,aver and vertical load F . Similarly to the PC girder-bridge with a straight tendon, a compressed beam of length L can consider the second-order shear effects in a simply supported girder-bridge post–tensioned by a parabolic tendon [Fig. 7 in Bonopera et al. (2021)]. G cm,aver is the time-dependent initial tangent shear modulus (Table 1). Consequently, the second-order small-deflection v tot ( a ) ( x ) can be obtained by assuming the transverse shear deformation and multiplying the first-order small-deflection v I,shear ( a ) ( x ) by the parameter of the second-order effects, k ( u ) [Eqs. (1)–(4) in Bonopera and De Matteis (2026)], where the effective flexural rigidity E cm,aver I 1 I , Nx =0 . I 1 I , Nx =0 = 1.44167 × 10 9 mm 4 is the effective cross-sectional second moment of the area of a simply supported PC girder-bridge according to the solution proposed by Song (Bonopera and Chang 2021). Likewise, the second-order small-deflection v tot,shear ( a ) ( x ) [Eq. (2) in Bonopera and De Matteis (2026)] can well be approximated by multiplying the first-order deflection v I,shear ( a ) ( x ) by the magnification factor of the second-order shear effects [1/(1– N x ,aver / N crE,shear,1 I , Nx =0 )], where the reference critical buckling load ( N crE,shear,1 I , Nx =0 =10,869 kN) is given as follows: