PSI - Issue 84

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

469

N

.

(1)

crE,1 , =0 crE,1 , =0 cm,aver 0 1 ( ) I Nx I Nx N G A +

N

=

crE,shear,1 , =0 I Nx

The reference Euler buckling load N crE,1 I , Nx =0 = π 2 E

cm,aver I 1 I , Nx =0 / L

2 . Notably that the approximated nonlinear deflected

shape v tot,shear ( a ) is attributable to that represented by the simply supported beam model depicted in Fig. 5 in Bonopera and Chang (2021). Accordingly, a three-point bending can also be exploited to identify the effective post–tensioning ( N a ,shear,3D FE ) in a concrete girder-bridge with parabolic tendon, subjected to time-dependent post–tensioning losses, using Eq. (1) reported in the first article. v tot,shear ( x ) ( x ) is the small-deflection measurement ( v i ). Vice versa, first-order buckling load ( N crE,shear,1,3D FE ) and deflection [ v I,shear,3D FE ( a ) ( x )] are numerical values which must be determined using a solid FE model of the PC girder-bridge. The initial tangent elastic modulus ( E cm,aver ) at the time of three-point bending must also be assumed (Table 1 and Section 3). More detailed information on the “static deflected shape” method for identifying long-term post–tensioning losses were reported in Bonopera and Chang (2021).

Table 2. Comparison between second-order small-deflections [ v tot,shear

( a ) ( x )] (“ Analytical – Shear ”) and measurements ( v i )

for each test combination and related to the layout depicted in Fig. 1 in Bonopera and Chang (2021).

Post– tensioning losses

Age of post– tensioning (days)

N 0 x 1

N 0 x 2

N 0 x ,aver

N x 1

N x 2

N x ,aver

F

v 1

v 2

v 3

v 4

v 5

v 6

v 7

(kN)

(kN)

(kN)

(%)

(kN)

(kN)

(kN)

(kN)

(mm) 1.02 1.08 1.18 1.48 1.57 1.76 1.83 1.94 2.18 1.06 1.12 1.22 1.44 1.52 1.68 1.87 1.98 2.22 2.25 2.38 2.63 1.16 1.23 1.36 1.48 1.56 1.76 1.83 1.93 2.18 2.26 2.39 2.69

(mm) 1.92 2.03 2.13 2.78 2.94 3.16 3.45 3.65 3.93 1.99 2.10 2.21 2.72 2.88 3.06 3.51 3.71 4.00 4.24 4.48 4.80 2.18 2.31 2.44 2.78 2.94 3.16 3.44 3.64 3.93 4.23 4.47 4.88

(mm) 2.56 2.71 2.79 3.70 3.91 4.15 4.59 4.85 5.16 2.64 2.79 2.89 3.62 3.83 4.01 4.68 4.95 5.24 5.65 5.97 6.33 2.91 3.08 3.21 3.71 3.92 4.16 4.59 4.85 5.20 5.65 5.97 6.45

(mm) 2.80 2.96 3.15 4.06 4.29 4.65 5.03 5.31 5.75 2.90 3.06 3.26 3.97 4.19 4.50 5.13 5.41 5.85 6.20 6.54 7.04 3.20 3.38 3.55 4.07 4.30 4.59 5.04 5.32 5.72 6.20 6.54 7.12

(mm) 2.56 2.70 2.86 3.70 3.90 4.25 4.59 4.84 5.27 2.64 2.78 2.97 3.61 3.80 4.11 4.67 4.92 5.36 5.65 5.95 6.45 2.91 3.07 3.27 3.71 3.91 4.22 4.59 4.84 5.26 5.65 5.95 6.53

(mm) 1.92 2.02 2.17 2.78 2.93 3.23 3.44 3.62 4.01 1.99 2.10 2.25 2.71 2.86 3.10 3.51 3.70 4.07 4.24 4.47 4.89 2.18 2.30 2.48 2.77 2.92 3.20 3.44 3.62 4.00 4.23 4.46 4.96

(mm) 1.02 1.07 1.26 1.48 1.56 1.87 1.83 1.93 2.33 1.06 1.12 1.29 1.45 1.53 1.78 1.87 1.97 2.35 2.26 2.38 2.81 1.16 1.22 1.43 1.47 1.55 1.82 1.83 1.93 2.25 2.25 2.37 2.78

Solid FE model (I-order) Analytical – Shear

23.2

Transducer

Solid FE model (I-order) Analytical – Shear

288

533

499

516

–14.0

533

499

516

34.2

Transducer

Solid FE model (I-order) Analytical – Shear

42.2

Transducer

Solid FE model (I-order) Analytical – Shear

24.2

Transducer

Solid FE model (I-order) Analytical – Shear

33.2

Transducer

290

533

499

516

–14.0

533

499

516

Solid FE model (I-order) Analytical – Shear

42.8

Transducer

Solid FE model (I-order) Analytical – Shear

51.4

Transducer

Solid FE model (I-order) Analytical – Shear

26.5

Transducer

Solid FE model (I-order) Analytical – Shear

34.1

Transducer

291

532

498

515

–14.2

532

498

515

Solid FE model (I-order) Analytical – Shear

42.1

Transducer

Solid FE model (I-order) Analytical – Shear

51.9

Transducer

5. Comparative assessment Table 2 listed the comparison of the small-deflections ( v i ) measured during three-point bending and after the occurrence of the time-dependent post–tensioning losses (Section 2). v i , at i = 1, …., 7, corresponded to the 288, 290 and 291 days of post–tensioning [Fig. 1 in Bonopera and Chang (2021)], were indeed compared with the small deflections [ v tot,shear ( a ) ( x )] calculated by the magnification factor formula, i.e., Eq. (5) reported in Bonopera and De