PSI - Issue 84

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

920

the 2D FE analyses were conducted in Strand7 (2010) by discretizing the concrete member into 16 Timoshenko beam elements [Fig. 2(b)]. Conversely, the straight tendons were discretized into 16 truss elements (with the absence of the tension axial forces). Notably, in the 2D FE model [Fig. 2(b)], the same loading configuration of the four-point bending was simulated (Fig. 1). Besides, the solution of the simply supported Timoshenko beam [ v I,shear ( a ) ( x )], loaded with two equidistant vertical forces ( F /2) [Eq. (1) in Bonopera and De Matteis 2026], provided with the same first-order deflection values. 4. Comparison of results The “load–displacement” and “load–strain” graphs for the specimen B2 showed a bilinear response, as visible in Fig. 7. In the first stage, an increment in loading was accompanied by a proportional increase in strain that corresponded to a reduction of pre–tensioning. Instead, during the second stage, an increase in applied load was no longer accompanied by a proportional increase in strain, as load was no longer transferred across the crack at the beam surface. The load at the transition from the first stage to the second one was taken as the “decompression loading”. To approximate the load causing the crack opening, the two linear branches of each curve were continued, and the intersection of the two lines was selected as the point at which the crack opened, as represented in graphs of Fig. 7.

Fig. 7. “Load-strain” and “load-displacement” graphs for the loading phase of the PC beam specimen (B2).

The “decompression load” was equal to 35 kN. The bending moment ( M ) corresponding to such a load was calculated according to the conventional beam theory and it was equal to 49.3 kNm. Furthermore, the bending moment due to the specimen’s self-weight ( M D ) was equal to 9.65 kNm, whilst the eccentricity ( e ) resulted in a value of 87.50 mm. Therefore, the effective pre–tensioning force in the strands, using the “cracking” method ( F PR = 428 kN), was determined by Eq. (1). Consequently, assuming that the design pre–tensioning force ( F PR,0 ) was of 475 kN, a percentage error of 9.9% in long-term pre–tensioning losses was gained. Notably, the design pre–tensioning ( F PR,0 ) was 8.5% of the first-order critical buckling load, N crE,shear,1,2D FE . The residual pre–tensioning forces ( N a ,shear,2D FE ) [Fig. 2(b)], identified through the “static deflected shape” method (Section 3.4), provided with values equal to 389 and 469 kN by respectively computing the deflection measurements v 3 = 5.57 and v 4 = 3.91 mm in Eq. (3). As a result, the percentage errors Δ 1 (with respect to F PR,0 ) were respectively −18.0 and −1.2%, whilst the percentage errors Δ 2 (with respect to F PR ) were respectively −9.0 and 9.7%. It is also worth noting that such residual pre –tensioning ( N a ,shear,2D FE ) were achieved [Eq. (3)] assuming an error in vertical load ( F = 25.0 kN) of −3 kN, i.e., equal to the error of −1% with respect to the load cell’s maximum capacity (= 300 kN). Thus, the vertical load was computed within Eq. (3) as follows: F = 22.0 kN. According to the comparison of the above findings, the residual pre–tensioning force was properly estimated through both approaches. The aforementioned work was limited to simply supported concrete girder-bridges with constant eccentricities of pre–tensioning steel reinforcements, thus enabling almost a complete isolation of the concrete block from the pre– tensioning forces. Hence, the validations above mentioned should be performed on more complex PC girder-bridges, e.g., statically indeterminate PC members by varying positions and shapes of the pre–tensioning reinforcements. 5. Conclusions This work compared a destructive and a non-destructive method to evaluate residual prestressing in concrete beams. The first approach was related to the “cracking” method. Conversely, the second was related to the “static deflected shape” one which, in turn, is based on the Timoshenko theory. Particularly, the static deflected shape of a simply