PSI - Issue 84

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

919

The second PC beam (B2) was instead tested adopting the “cracking” method. The first step was to load the member overcoming its elastic limit until the occurrence of a series of flexural cracks. According to the configuration represented in Fig. 5(b), the electro-optical strain measurement (ESR) sensors indicated that the crack patterns were located close to the instrumentation and affected the ESR curves with a rapid change of trend, as shown in Fig. 5(a).

(a) (b) Fig. 5. (a) Load-strain relation for “cracking” method; (b) Instruments for strain and displacement measurement.

The cracks were underlined with a marker to guarantee their accurate localization after the PC beam was unloaded. Particularly, five cracks were depicted at the constant bending moment area, as shown in Fig. 6(a). Notably, four cracks were instrumented using two TR and two ESR sensors. The PC beam was then reloaded and the “load-strain” graphs were shown for each instrument located above the cracks, as depicted in Fig. 6(b).

(a)

(b) Fig. 6. (a) PC beam specimen (B2) during testing; (b) “Load-strain” and “load-displacement” graphs for the loading phase. 3.4. Application of the non-destructive method (“static deflected shape” method) The residual pre–tensioning force ( N a ,shear,2D FE ) in the beam “specimen B2” was estimated through the “static deflected shape” method proposed by Bonopera and De Matteis (2026) and according to the considerations about the use of the magnification factor formula including shear deformation [Eq. (2)] (Section 2.2). The vertical load ( F ) during four-point bending, and measured by the 300 kN load cell (located between the actuator and the steel frame) was gradually incremented to ≈ 25.0 kN. Subsequently, two analytical values of the residual pre–tensioning force ( N a ,shear,2D FE ) were determined [Eq. (3)] assuming the experimental parameters of chord elastic modulus ( E = 45.0 GPa), shear modulus ( G = 18.75 GPa), midspan small-deflection measurement ( v 3 ) and 3/4 small-deflection one ( v 4 ). In detail, such analytical values ( N a ,shear,2D FE ) were compared with the design pre–tensioning force ( F PR,0 = 475 kN) of specimen B2 and the corresponding residual pre–tensioning identified through the “cracking” method ( F PR = 428 kN). The related comparisons between such pre– tensioning forces were expressed by the percentage errors Δ 1 = ( N a ,shear,2D FE − F PR,0 ) / F PR,0 and Δ 2 = ( N a ,shear,2D FE − F PR ) / F PR . Particularly, the deflection measurements ( v 3 and v 4 ) were respectively computed as parameters for the small-deflection v tot,shear ( x ) ( x ) [Eq. (3)]. Vice versa, the first-order critical buckling load of specimen B2 ( N crE,shear,1,2D FE = 5,559 kN) and the corresponding first-order deflections v I,shear,2D FE ( a ) ( x ) were determined using the reference 2D FE model proposed by Jaiswal (2008) [Fig. 2(b)]. Indeed, the behavior of a PC girder-bridge can properly be described by considering an equivalent beam–tendon system (Bonopera and Chang 2021). In the 2D FE model, when a pre–tensioned concrete girder-bridge is elaborated, beam and tendon elements contain common nodes connected by rigid links, as represented in Fig. 2(b). The small eccentricities along the specimen B2 are the distance of the straight tendons from the centroid of the beam cross-section [Fig. 3(a)]. Moreover,