PSI - Issue 84

Andrea Nino Consiglio et al. / Procedia Structural Integrity 84 (2026) 914–921

917

subsequently to a vertical load ( F ) at its midspan. The prestressing ( N ) is assumed to be externally applied at the beam ends. The concrete chord elastic modulus ( E ), cross-sectional area ( A ) and second moment of the area ( I ) are known parameters. v (0) is the initial deflected shape of the PC beam which occurs after the application of prestressing ( N ). Conversely, v tot ( a ) is the deflected shape under the application of prestressing N and vertical load F [Fig. 1(c) in Bonopera et al. 2018]. It can be expressed by Eqs. (3a) and (3b) reported in Bonopera et al. (2018). Notably, a compressed beam of length L can consider the second-order shear effects in a simply supported concrete girder-bridge pre–tensioned by a straight tendon [Fig. 1(a) in Bonopera et al. 2018]. This configuration characterized the experiments illustrated in Section 3. Shear modulus G = E / [2(1+  )], where Poisson’s ratio  =0.2. Accordingly, the small-deflection v tot ( a ) ( x ) can be provided assuming the transverse shear deformability (Timoshenko theory) and multiplying the first-order 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]. The 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 / N crE,shear,1 ) (Bažant and Cedolin 2010; Bonopera and De Matteis 2026) as reported as follows: ( ) I, shear, 2D FE ( ) tot, shear crE,shear,1,2D FE ( ) ( ) 1 a x v x v x N N = − (2) A static bending test with an applied pre–tensioned force ( N ) and an additional vertical load ( F ), positioned at a cross section/s, can be executed to measure the small-deflection, v tot,shear ( x ) ( x ), along a concrete girder-bridge [Fig. 1(c) in Bonopera et al. 2018]. Consequently, the formulation of the magnification factor including shear deformation can be utilized to evaluate the pre–tensioning ( N a ,shear,2D FE ) through equation (3), and according to the “static deflected shape” method proposed by Bonopera and De Matteis (2026). Indeed, static small-deflections were exploited as parameters for evaluating residual prestressing because the corresponding vibrational frequencies are not significantly influenced by the prestressing variations (Bonopera and Chang 2021; Gandelli et al. 2024). (3) application of a vertical load ( F ); (2) Obtain the first-order critical buckling load (including shear deformation), N crE,shear,1,2D FE , of the pre–tensioned concrete girder-bridge using a Two-Dimensional (2D) Finite-Element (FE) model [Fig. 2(b)]; (3) Solve equation (3) for identifying residual pre–tensioning ( N a ,shear,2D FE ) by determining the first-order deflection [ v I,shear, 2D FE ( a ) ( x )] through the same 2D FE model [Fig. 2(b)]. More detailed information on the procedures of the non-destructive method, both small-deflection measurements and analytical ones were reported in Bonopera et al. (2018) and Bonopera and De Matteis (2026). 3. Experimental campaign 3.1. Pre–tensioned concrete beam specimens A set of simply supported concrete beams were used to investigate the evaluation of residual pre–tensioning within the framework of an experimental program conducted at the company “Precompressi Centro Nord” (Novara, Italy). Particularly, pre–tensioned concrete beams cast in 2003 were tested. Such PC beam specimens were selected based on their geometry because representative of typical pre-cast slabs for road tunnels in Italy. Their length ( L TOT ) was 7,220 mm, whilst their cross-sectional area values were reported in Fig. 3(a). Their relevant geometric and material properties were instead listed in Table 1. Two pinned-end restraints were positioned at their ends for a span L = 7,000 mm. As a result, their slenderness ratio was 70, whereas their length/height ( L /h) ratio was 23. Each member was pre–tensioned by seven-wire mono- strand straight tendons, 1/2’’diameter, and one wire mono -strand straight tendon, 5 mm diameter. Moreover, both PC beams did not have any longitudinal or transversal steel reinforcement. The concrete compressive strength ( f c ) and chord elastic modulus ( E ) of both PC beam specimens were estimated through the Sonreb test, according to the UNI EN 12504 (CEN 2021). This test combines ultrasonic and rebound method to evaluate the mechanical properties of concrete which, in turn, is affected by its consolidation/hardening Generally, the residual pre–tensioning identification must be carried on by the following steps: (1) Measure a small deflection [ v tot,shear ( x ) ( x )] along the pre–tensioned concrete girder-bridge under investigation, and following the