PSI - Issue 84

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

461

( E aver ) and geometric properties ( A and I ) are known values. v which occurs after the application of prestressing ( N ). Contrariwise, v tot ( a ) is the nonlinear deflected shape under the application of prestressing N and vertical load F . A compressed member of length L can consider the second-order shear effects in a simply supported girder-bridge post–tensioned by a straight tendon [Fig. 1(a) in Bonopera et al. (2018)]. This configuration characterized the three-point bending tests presented in Section 2. Shear modulus G aver = E aver / [2 (1+ υ )]. Poisson’s ratio υ = 0.2. Therefore, the second-order small-deflection v tot ( a ) ( x ) can be achieved by assuming the transverse shear deformation and multiplying the first-order small-deflection v I,shear ( a ) ( x ) [Eq. (1) in Bonopera and De Matteis (2026)] by the parameter of the second-order effects, k ( u ) [Eqs. (2–4) in Bonopera and De Matteis (2026)]. Yet, the second-order small-deflection v tot,shear ( a ) ( x ) [Eq. (2) in Bonopera and De Matteis (2026)] can properly be expressed by multiplying the first-order deflection v I,shear ( a ) ( x ) by the magnification factor of the second order shear effects, 1/(1– N / N crE,shear,1 ), as showed by Eq. (5). Accordingly, a three-point bending experiment with an assigned post–tensioning ( N ) and an additional vertical load ( F ) can be conducted to measure the second-order small deflection, v tot,shear ( x ) ( x ), along a concrete girder-bridge [Fig. 1(c) in Bonopera et al. (2018)]. Consequently, based on the “static deflected shape” method revised by Bonopera and De Matteis (2026), the formulation of the magnification factor assuming shear deformation can be adopted to identify the existing post–tensioning ( N a ,shear,3D FE ) as follows: (0) is the initial nonlinear camber of the PC girder-bridge, Generally, the identification of the effective post–tensioning ( N a ,shear,3D FE ) must be done through the following phases: (1) Measure a small-deflection along the PC girder-bridge under investigation [ v tot,shear ( x ) ( x )] by applying a vertical load ( F ); (2) Obtain the first-order critical buckling load including shear deformation of the PC girder-bridge ( N crE,shear,1,3D FE ) using a solid FE model (Section 3); (3) Resolve Eq. (1) to identify the post–tensioning ( N a ,shear,3D FE ) by determining the first-order small-deflection [ v I,shear,3D FE ( a ) ( x )] using the same PC girder-bridge’s FE model. More detailed information on the nondestructive method’s procedures and analytical/numerical deflections were reported in Bonopera et al. (2018) and Bonopera and De Matteis (2026). 5. Comparative assessment Table 2 had the main goal to itemize the comparison of the small-deflection measurements ( v i ) from the different test combinations of the three-point bending (Section 2). Particularly, v i , at i = 1, …., 7 [ Fig. 5 in Bonopera et al. (2018)], were compared with the corresponding second-order small-deflections [ v tot,shear,3D FE ( a ) ( x )] achieved from the solid FE modeling [Fig. 1(a)] (Section 3). The related average time-dependent elastic moduli ( E aver and G aver ), applied post–tensioning ( N 0 and N x ) and vertical loads ( F ) were respectively accounted for in Strand7 (2010). Besides, in Table 2, the second-order small-deflections denoted as “ Analytical – Shear ” were the displacements determined by the magnification factor formula of the second-order shear effects (Section 4), similarly itemized in Table 1 in Bonopera and De Matteis (2026). Thus, comparing the distinct displacement values, the absolute average error between second-order small-deflections [ v tot,shear,3D FE ( a ) ( x )] and small-deflection measurements ( v i ) was 0.09 mm, corresponding to an average percentage error equal to 4.2%. Considering the specimen’s cross-sections at i = 2 and 3, a maximum percentage error of 9.2% was instead gained within the 9 test combinations. Contrariwise, when the magnification factor formula was assumed [Eq. (5) in Bonopera and De Matteis (2026)], the absolute average error between second-order small-deflections [ v tot,shear ( a ) ( x )] and deflection measurements ( v i ) was 0.04 mm, corresponding to an average percentage error of −1.5%. Vice versa, assuming the cross -sections at i = 2 and 3, a maximum percentage error of −3.3% was obtained among th e test combinations. Notably, the aforementioned comparison between small deflections [ v tot,shear,3D FE ( a ) ( x )] and deflection measurements ( v i ), at i = 1, …., 7 , exhibited systematic errors due to the ineffective mesh size resolutions adopted within the solid FE modeling of the PC girder-bridge specimen [Fig. 1(a)]. Yet, the above mentioned findings still matched with the outcomes regarding (axially unloaded) simply supported members with a rectangular cross-section, in which the additional shearing force effect on the first-order deflections [ v I ( a ) ( x )] is ≈ 4% for a L /h ratio = 10 (Timoshenko 1946). More detailed information on the small-deflection measurements ( v i ) were illustrated in Bonopera et al. (2018) and Bonopera and De Matteis (2026). ( ) I, shear, 3D FE a ,shear,3D FE crE,shear,1,3D FE ( ) tot, shear x ( ) x 1  − ( ) x a v N N v        = . (1)